Hey there. I am planning of taking a real analysis course but I'm so scared of the rigorous mathematical maturity involved in it. I am planning to get a leg up by looking at the material and seeing what I have to know. What are the main proofs techniques I can get a hang of in order to succeed in the course? I have only taken two proofs courses but I did kind of bad but want to get better.
Hi there, i find the book online. I haven't purchase yet; but before i do i want your opinion whether or not is it saved to take the course as independent study???? Thanks.....
@@youssephfofana9226 Analysis is a course worth taking from all possible sources. All your life. There's never can be "enough" of this only "true" mathematics.
Just took a real analysis exam with 20 proofs, 5 definitions, 10 T/F, 5 open-ended questions, and 3 T/F questions where you have to prove your answer. In an hour. Dropping out now. This was my last math course... I knew all the material, there just wasn't nearly enough time to write everything. Edit: I didn't drop out. After commenting this, I got angry because the prof said he didn't want me to pass so I studied hard and memorized every proof from 4 chapters of the textbook to do well on the final. Passed the course and I'm gonna graduate now.
@@ToddlerAnnihilator666 From what I've seen from some online math groups, there seems to be a stereotype of certain areas and their math faculty, I couldn't elaborate too much as I don't quite understand it or have the experience to speak on it's validity
@@soupy5890 Ok you are indeed correct. I was lucky to attend uni with fairly down to earth staff/professors. But those that transfered from other unis say that to keep their employment rates (and other statistics) higher they purposely fail students after certain qouta has been met
The more practice with mock exam questions, the more your confidence grows. I frequently visit Mathematics Stackoverflow and MathOverflow, considerably the best hubs where I view tough exam-like questions and others' solutions. I learn much by brain-picking gifted mathematicians there. Don't mind me, as I am only an intermediate in mathematics.
The video is quite helpful. Thanks for the suggestions. Struggling with the real analysis right now. Questions are not super hard, but I'm nowhere near solving them. It's frustrating that after so many hours of hard-working, I still have a hard time writing down proper proof without oversimplifying or complicating things. Whereas my classmates are gifted, discussing recondite ideas with prof all the time.
I'm actually kind of surprised to hear that (at least some) math programs use real analysis as the first proof-based course. At my school, they used the discrete math as the "intro to proof" course. As a computer science major, it was hard enough for me to pass this class when we were writing proofs about sets; I can't imagine having to throw calculus into the mix!
Agree on repeatedly failing and trying then finally getting it. That's probably the best thing that I gotten out of being a math major. There's no way I'm giving up on anything after completing my degree.
I totally feel you. The key here is really not giving up. I am taking analysis right now and it’s extremely difficult to me. But I still force myself to try again and again.
Killer in what way? For context I'm considering taking Real Analysis as my elective (mainly out of curiosity and for the challenge) but I've heard often that Topology helps a lot with understanding real analysis and should be taken before analysis.
@@FsimulatorX If the presentation is just abstact and formal, then you might ask 'what is all this, and what are the practical examples that lead to these definitions?' For example, the Euclidean space R^n is a prototype for a toplogical space. A well motivated book is Topology now! by Messer".
I came here after crying my eyes out because of a real analysis assignment. I really wish I knew this beforehand, it’s so disheartening to be in a class where you feel so inadequate
Here's a nugget: Take Set Theory if it's offered or study it on your own. This is like the "introduction" to Real Analysis. This will get you thinking about the abstract logic when studying real numbers and knowing how to "speak the real analysis language" when it's time to discuss integration and differentiation utilizing sets because you're not going to be computing hardly anything, but writing nothing, but proofs and more proofs
My university does not offer a set theory course but it does offer an Intro to mathematical concepts/proofs course that covers logic, proofs, sets, functions, relations and number theory.
This subject really shook me in my first sem class. It takes so much time to absorb, and is just So different. Watching your video's making me realize how students first upon entering college-level math need to be given intro transition classes into each subject each semester. Like, u need some storytelling and constant context to get through that!
@@youssephfofana9226 I had a skinny little (80 pgs or so) book called Set Theory that was used in class. Sry can't remember the author but it might've been my prof (last name Bradley). Very concise explanations on axioms, (well) ordering, proof by induction, etc
I will be teaching Real Analysis this semester. In preparation to the semester, I was planning to find a tasteful video on the history of RA. Then instead I found this video! These things came to me naturally when I was once a student, so, I was blind to most of the issues. But by experience I know, these are exactly where my students may struggle. I will share this video with my students. They'll surely appreciate it. Thanks for the excellent video!
Being an undergraduate engineering student I didn't have an acquaintance with RA. I was really good at calculus, but RA kicked my butt when I got into grad school. I needed to strengthen my pure mathematics background for what I was into. And as you mentioned, it is NOT calculus. So I had trouble with the abstract nature of it. It still bugs me after all these years! And that is why I watched your video. Thanks.
Real analysis was by far the most troubling undergrad math course I've taken as a math major. Don't know why I still have the textbook I don't understand 95% of the text lol
You can do it! .. It IS hard to digest, but get yourself s many resources you can to understand how to deal with all the elements that make up the subject. Best wishes!
As someone attempting to self study real analysis, this is so useful! This entire video managed to put into words what I’ve felt for the past month. It’s been really hard to even do the most basic proofs, but this has given me some jump off points to speed up progress. Thank you.
1) The real Analysis course is nothing like a Calculus course. It's logistically rigorous and proof-based. It's not taking derivatives, factoring, plugging, etc. 2) Be familiar with some proof techniques, such as mathematical induction. 3) Be extremely familiar with definitions: definition matters. You need definitions to start a logistic proof that your professor like. 4) Write down the definitions you know and write down what you want to prove (the conditions and the conclusions). It can be a great way to trigger a correct proof. 5) Be familiar with logistic quantifiers, all kinds of notations. 6) Persistence is key. You have to stick to learning and trust yourself. Never giving up is important to nail this class.
It’s funny how big TH-cam is. You are clearly doing a great job, you have videos that are well made and well targeted and raking in the views. This is the first video I’ve seen of yours. Keep up the great work!
Great video. Here's what you must know before becoming a math major. Real Analysis (aka 'Advanced Calculus') is usually the first proof-based course in college and it isn't normally taken until FAR TOO LATE to realize you don't understand proofs at all and you may NEVER be able to handle proof-based courses, making a math degree effectively unattainable for you. Real Analysis is usually taken in your junior year and these proof-based courses make up your last two years of college yet they are a COMPLETE CHANGE OF DIRECTION from what you thought math was all about. It's almost criminal that this isn't revealed and emphasized EARLY ON in college, but it is not. Math profs at university have pure math PhDs and were born knowing how to do this stuff (no kidding). They assume you are the same. College advising sucks and no one will tell you the following. Being good at 'plug 'n chug' will not help you with proofs. You can be an A+ student in all calculus courses and fail proof-based courses such as Real Analysis. If you happen to be reading this before it's too late, get Laura Alcock's book 'How to Think About Analysis' and STUDY IT CAREFULLY. If you can't follow her material, you are probably in the wrong major. Her presentation is, by far, the best I've seen. Even applied math majors will normally get a big dose of proof-based courses that may stop them - dead in their tracks. Get lots of help or change majors, NOW! Asking the prof for help is a waste of time. Get online and find someone on Wyzant or something like that who is good at pure math and do one-on-one tutoring. But, good luck! If you can't do proofs, a Math PhD is not going to happen. So, even if you complete an Applied Math program, you better have computer science skills in order to get a job in a relevant field (except teaching public school). You won't be teaching at University without a PhD in math.
@@xHannaHx33 What book are you using and what section are you lost on? If I can find the book online, I'll take a look at the section. Remember, get help ASAP. Don't wait. Perhaps try Wyzant. Upper level math is not commonly found on TH-cam. If this is your first 'proofs' class, it will make or break you.
oh, I started real analysis book by myself just bc I found it interresting to learn something abstract myself PS: i am electrical engineer at high school. Nearly every question seems unsolvable while deffinitions are so simple and logic. I really felt stupid bc I couldnt proof such clear things like intersection with such simple deffinitions. But now I see that it's not that simpel as I though))))
I hated that I didn't know proof as well as I needed, so I took a break, studied a HUGE amount of proof-related material. It's an advantage you have as an older student where you don't care how long it takes to graduate. My goal was simply to not complete a course/class unless I considered myself to have mastered the material.
Yes, yes, his point number 4! Sometimes, just start "Since, (definition) and (definition), then because of .... we see that (conclusion.)" Write this in "math" to practice. Trust me, you will be on your way.
What's book do you use for introduction proof??? The name and the author precisely the ISBN# will be great. I'm looking forward as independent study. Thanks...
@@youssephfofana9226 hmm I don't have the ISBN, but it's called "Analysis with an Introduction to Proof" by Steven Lay. Here is the link: www.pearson.com/store/p/analysis-with-an-introduction-to-proof/P100001370585/9780321747471
2:38 induction definitions. alarm bells. I remember when I was sitting in an upper-division class and remember the panic when the professor would right Defn. - Now I understand. LOL 5:20 Start at what You know and The answer you want, meet the 2. 6:30 Notations, quantifiers, symbols,
Currently doing a major in applied math and computer science and Real Analysis was one of the most dreaded courses. Unfortunately got stuck with Real Analysis, Complex Analysis and Stochastic Processes all in one semester haha. But who doesn't love a good challenge ;)
lol how did it go man? I'm thinking of taking real analysis, abstract algebra and stochastic processes on the same semester. Complex Analysis sounds dreadfully worse though XD
Real Analysis as the FIRST proofs-based course? That's insane, I had my own introduction to proofs class and apparently talking to some computer science students my introductory class was three discrete math classes combined. Then I picked up a Real Analysis book to see what it's like and the first Unit just casually summarized everything I learned in that class. I'm still trying to read the book; it's intense and the exercises are brutal, but I find RA VERY rewarding. I was forced to switch Real Analysis with Abstract Algebra and Probability Proofs for the upcoming year, but hopefully I can take RA soon.
I agree with you about not giving up and persisting. Sometimes the first encounter with some new material is like hitting an impenetrable wall. You have to persist in battering that wall, until it gives way to your understanding.
@@BriTheMathGuy When I first tried abstract algebra I saw the material and just about had a panic attack. I am not ready for it yet. I am brushing up on my fundamentals first before I try again.
All of this is why they should teach Real Analysis *before* Calculus. If you're balancing a checkbook, you don't need Calculus. However, if you're using Calculus, you need to understand the tools you're using.
so glad i was able to take geometry and abstract algebra before RA. i had never seen non-E geo before then, so that solidified the importance of definitions. the set theory and structural concepts of ab. alg. helped me get my mind right. RA was still a challenge, but it’s like i was looking at calculus from “the other side,” so i really felt like i was going somewhere in my math understanding
I watched this video before starting the real analysis course so I couldn't understand what he was talking about back then. Now, 2 months later, I can understand and relate to each and everything he is saying. I followed what he told and I can say I am starting to get a grip, still very very very far from the end but I am moving atleast. Edit: I think I'm gonna fail this semester 😭
My God, what you just said here about the essential importance of knowing definitions really opens my eyes to so much, not just in math, but in life and the world in general...
Issue: What is a differential of an irrational argument? Let a= some rational approximation, and A be the irrational number itself (if that makes sense). Then A - a > dA and there is no way a + dA > A
Re. Real Analysis vs Calculus sequence: In some teaching sequences, Real Analysis is taught _before_ Calculus. There is a certain logic in doing so, theory before application. Hence why Calculus was usually feared being the final class taken. In the most of US (United States) for the last 40 (or 50?) years, that is reversed because more practical teaching that way. In Mathematics Departments' defense, they are teaching to a much larger audience than potential math majors; Chemistry, Physics, pre-med, econ, engineering, Comp Sci, and possibly any other are all required having mathematics, and in most cases, Calculus and Differential Equations. So makes much more sense that the first year(s) courses are calculus then switch to the more rigorous theoretical framework of upper division (junior & senior level) classes which are geared specific toward math majors. Unless one is doing Harvard's Math 55, they are not likely to study Real Analysis in their first year.
It annoyed me into oblivion that in my math class I had to do "proofs" that weren't proofs - more like a basic vague sketch of ideas that really has no structure - at all and it was almost like the teachers were actively trying to persuade us that proofs don't exist. I can't even talk to my classmates now, because just a slight derail from the beaten path of plug and chugging leaves them confused, and that's kinda sad, being unable to really share these great ideas with anyone as an EECS undergrad(At least I hope that's the reason) :D It's a lot of work to learn this properly though
This is so true! I'm in my final year as an Electronics undergrad and in most of our classes that involve maths to some extent, the teachers really don't care about rigour and we just take most things for granted...
Having accomplished 2 Math Degrees so far...I highly recommend a std Web Search for: 'real analysis prerequisite courses', reason being that there are a couple of basic lower-level Math courses which WILL help greatly such as: *Discrete Math (emphasis on Logic & Induction, Boolean Algebra, etc) *Abstract Algebra I/II (sometimes called Mathematical Methods in undergrad programmes) *Introductory Analysis (basic methods of proof, etc). However, apart from these there are the more 'specialized' courses route such as: *Advanced Linear Algebra *Set Theory *Point-Set Topology In hindsight, I presonally summarize Real Analysis as (Advanced) Calculus I-III using R.A. methods & similarly for my personal Major & all-time favorite topic 'Complex Analysis,' Calculus I-IV using C.A. methods. Footnote: make no mistake, these advanced 'Analysis' type courses require committment, dedication & HARD WORK to SUCCEED! Be prepared to FAIL (& YES I failed, did a few repeats along the way), but KEEP practising as commented in this Video. You'll eventually appreciate the sheer beauty, mystery & wonder of the mathematical Universe...as I have. So, good luck on your journeys, my mathematical friends!!
Interesting. I see how it would be difficult to unlearn Calculus from the geometric approach as it is introduced and developed in the usual way, then relearn it from an arithmetic/algbraic/logic point of view. *Me:* "I understand Calculus." *ANALYSIS:* "You don't understand the Number Line."
One small thing I would like to add too is the use of counterexamples. I didn't fully understand this at the time when I took Real. The idea that for some proofs (not all) you can use a counter example that contradicts the claim. It's easy to overthink this as it's hard to think of an example that contradicts the claim.
One of the most difficult thing for me as a beginner is to use mathematical definitions to proof certain theorems...these theorems makes you realize that you don't know the actual meaning (essence) of the topic.. 😑😑
I couldn't do it in university - it was my 8th class, I had 7 other courses. The pressure was too much so I dropped it and the idea of getting a double major in math and engineering. Without the pressure I now use it to escape the insanity around me and find it very interesting - lots of little puzzles but no harder than puzzles you would find in a book of puzzles. Of course, a book of puzzles is a waste of time since you can get the same pleasure and learn something practical and useful like real analysis. If you take your time, it's not that hard - stop at something you don't understand until you do - if that takes a week, no big deal. I don't know why anyone would solve puzzles when a desire to be challenged can be met with so many useful subjects like philosophy and math and these really develop your mind. You know it's working when you start to find sitcoms and movies boring. I'm 56 and can still learn this kind of stuff although it is harder than when I was in my twenties.
It means that for every epsilon greater than 0, there exists an n in the Natural Numbers such that for all n greater than some k (also a natural number), then the absolute value of the sequence (X_n) is less than epsilon. There!
I had a master of science degree in mechanical engineering and even had taken two graduate course in Classical Mechanics and Mathematical Physics when I decided I wanted to get a masters degree in mathematics. Real Analysis was the first course. I was blown away and had to drop it. I wish this video had been around all those years ago! It was like approaching math in a completely different way. I just couldn't wrap my head around it.
Real Analysis was difficult for me as an undergraduate at UCSC…fact is, I didn’t pass the first time. The next year I met Ralph Abraham who was teaching at UCSC at the time (ca. 1982). He said that there were two ways to understand mathematics…. One was the brute force method in which you struggle for hours on end to understand the problem. He said that this method will be difficult, but can certainly succeed. The other, is to think about the problem before you go to sleep at night… If upon awakening the next morning, you have the answer to the problem then you will know that the path of mathematics is the right path for you. I asked why this was…he said that much of mathematics lies in the dream world. For example, the idea of complex numbers only makes sense in the dream-world. He said that this was the only way to truly understand Mathematics. And it does work…I could not have received a degree in Mathematics otherwise. RIP my friend.
Do you have any tips on how to (improve) remember(ing) math theorems and definitions? Most of the times I sort of get the point of how you can use it and understand what it does, yet I always struggle to memorize the more precise/small parts like for example some given requirements, preconditions or exceptions.
Thanks Bri! What you're providing are the axioms to have when you walk through the classroom door. That way, Real Analysis axioms are the second set of assumptions you learn instead of the first!
I studied Measure Theory without any knowledge about Real Analysis. For 3 months up until now I studied every day, starting from the definitions of open sets, etc. I still remember the first several classes I could not understand what prof said. So I just quitted the class and studied it myself. Now I can understand the proofs and the solutions of exercises, but I feel like I’m still far away from solving exercises 😂. Tomorrow I have exam. Wish myself luck 😅
@@gauravbharwan6377 Yes! 16/20. I came back right after I had result. Turn out one (over 3) exam question was similar to one exercise and I learnt it by heart, so... well
Real analysis is about the core nature of real numbers.I personally feel the reason why students find it difficult is that they study functions,limits ,differentiation ,integration and sequences intuitively. Most students do not have a sound knowledge of set theory and logic and therefore lack rigor in their veins.
Agree with this completely. I was just thrown into real analysis (it was my very first lecture in university) and had to painfully realise this myself as the lecturer didn't say it.
Gustavo López much appreciated! I just started using Bartle and Sherbert's Introduction to Real Analysis or something. What's the title of Terry's book?
@@marks4982 Analysis 1 and Analysis 2. However, if you are in high school I recommend " A story of real analysis". It covers a typical first course in analysis with a lens of history describing how the subject evolved and providing motivations for dry definitions. The mathematical maturity you'll develop after completing this book will be great. I also used this in high school. You need to know differential and integral calculus ( because it treats examples that way) and some basic differential equations.
I read and went through all the questions from Richard hammock's book of proof. I don't know if this is the same this as pure logic, but it is beautiful. To me, real analysis sounds like it builds on the bones of logic. But what kind of course or subject lies at a deeper more fundamental level. Like the three rules of thought and a proof for the existence of truth.
I studied real analysis with the late Prof. Rudolf Vyborny, a Czech emigre, in the 1980_s . It was one of the most intellectually fulfilling studies in my life.; but it is hard and challenging for sure. I would encourage anyone who feels a bit overwhelmed by the rigour of the subject to persist and to believe that they will get there in the end.
Velleman's How to Prove It opened the world of math to me. I thought I'd need proofs for Calc 1, but aside from delta/epsilon🤷♂️ Everyone should be obliged to take a course based on Velleman before embarking on a math major. He goes thru symbolic logic, quantifiers, everything.
I took Real Analysis in the early 80's and I wish I had this video to watch back then before I took it. I honestly was not ready for it and the prof (who wrote the textbook) never really gave any overview on the topic or discussed any of the techniques required to do the proofs. He kind of assumed we knew how to do "Proof by Contradiction" or "Proof by Induction" etc. In fact, one of our assignments was to prove the Dedekind Cut Theorem. I think everyone got lost on that one.
Lol watching this one day after my real analysis exam has me like, wow....CAN DEFINITELY RELATE!!! By the way, you're the one who taught me what a topology is. Absolutely love your video on that, it was extremely helpful and I love this video. It's ℝeal lol. It's sorta sad the way the math program is made though, before this level, no one will know what's about to hit them. I only knew that everyone said they "hate" real analysis. Never heard nor saw anything else about it. Now I understand that it's an analysis based course, which I kinda like tbh, and kinda hate. I loveeeeee finding solutions to problems lol it makes me feel like a jackpot winner LOL, but on the other hand, I HATEEEEE all of these seemingly random, yet connected and absolutely necessary theorems and definitions. I stopped counting them when we hit topology. LOL. I do however have a very LONGGGGGG list of almost all of the definitions and theorems stockpiled into a MS Word document hahaha... Crazy stuff. Tbh, Advanced Calculus was much harder for me than ℝeal analysis because at least by the time I hit ℝeal analysis, I was already exposed to a great many of the definitions that we would need to use.
I’m taking real analysis this next year, and I feel relatively good with the things you mentioned since I’ve already taken an intro proofs course, and 2 abstract algebra courses. The only thing that worries me is the fact that I haven’t taken any type of calculus in 2 years. But nevertheless I still feel confident.
you don't really need to 'know' calculus, but it does help knowing calculus so that the motivation behind learning analysis is there. When you know the end goal, you gain a greater appreciation for the machinery they build up in the class.
If people knew what lie in store for them, they would pay more attention in calculus to the proofs of the theorems, not just the applications. I remember I had the "I just want to know how to do this" mentality. As Brian mentions in his videos, you are in for a rude awakening if you do this once you get to real analysis.
About the sandwiching "technique" that is mentioned here: I remember using it in Euclidean Geometry in my early teens. Then I considered it natural so that everyone should be using it. I tried to teach it to one student two years my junior, but it seemed easier said than practiced for him.
In my current undergrad experience its interesting the courses which definetly are built upon complicated niche rigorous proofs are often the classes where you arent required to do those proofs, because many students who find calculus appealing end up reaching into areas like differential geometry and geometric calculus those courses are often taught at the undergrad levels with out too many proofs even tho there “past” calculus they require real analysis to work
Great video, I graduated from my math BA program in January and many concepts that were still fuzzy are finally starting to click now and it happens at the most random times so I would agree that persistence is key, I don’t think I would have done well in Real Analysis if I had not taken a proof course first. Set theory is your friend 🙂
Before taking the class, all this advice sounded obvious. After taking the class however, all this advice sounds painfully personal. Suffice to say i had to go over all the proof-based classes again after finishing the first year.
In days past, there wasn't a single "proof" class offered or available. You took the 3 core Calc classes, an ODE & Linear Alg. class then, BOOM you're were in a Real Analysis using baby Rudin's book. Professors assumed you knew how to prove things and it was sink or swim. There were very few students that made it. I guess the Math Dept. used this as one of their "weeding out" courses.
The divide between school level calculus and undergraduate calculus/analysis applies to pretty much all subjects of mathematics. You think arithmetic is easy? Wait till you learn number theory. You think polynomials and theory of equations are easy? Wait till you learn abstract algebra. You think linear algebra is easy? Wait till you learn linear algebra, (but done right). And of course finally: You thought baby Rudin was hard. Just wait till you read papa Rudin.
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What I Wish I knew before becoming a Math Major!
th-cam.com/video/wk28BSaLszo/w-d-xo.html
Hey there. I am planning of taking a real analysis course but I'm so scared of the rigorous mathematical maturity involved in it. I am planning to get a leg up by looking at the material and seeing what I have to know. What are the main proofs techniques I can get a hang of in order to succeed in the course? I have only taken two proofs courses but I did kind of bad but want to get better.
Hi there, i find the book online. I haven't purchase yet; but before i do i want your opinion whether or not is it saved to take the course as independent study????
Thanks.....
@@youssephfofana9226 Analysis is a course worth taking from all possible sources. All your life. There's never can be "enough" of this only "true" mathematics.
@@briancannard7335 Thanks, good to know. I Wii immerse myself in different sources...
So what you do if someone has the problems you are referring to the start of the video how you overcome it?
Freshman Math Major: Calculus is cool
Real Analysis: I’m gonna end this man’s whole career
So true 😅
Man I’m taking astrophysics in university of toronto this is my first year and I took real analysis and man… I need help💀
I feel like this converges onto me.
@@priyanshugoel3030 uniformly.....and absolutely...
functional analysis and pde: im gonna end this man's life
Berkeley's Real Analysis class taught by Dmitry Vaintrob, just assigned this video as an introductory assignment haha!
Wow really?
@@BriTheMathGuy yeah haha, great channel!
@@suryaprakashvengadesan4930 Thanks very much!
Wish I had tour professor. My professor in real analysis didn’t know how email worked
@@walkerscoral damn, he must be a sad guy
This guy makes the first comprehensive beginner's perspective on real analysis that I've seen one semester AFTER I took it!
😅 i completed my masters in mathematics
Just took a real analysis exam with 20 proofs, 5 definitions, 10 T/F, 5 open-ended questions, and 3 T/F questions where you have to prove your answer. In an hour. Dropping out now. This was my last math course...
I knew all the material, there just wasn't nearly enough time to write everything.
Edit: I didn't drop out. After commenting this, I got angry because the prof said he didn't want me to pass so I studied hard and memorized every proof from 4 chapters of the textbook to do well on the final. Passed the course and I'm gonna graduate now.
What kind of professor would tell you they don't want you to pass? Are you in eastern Europe or something?
@@FsimulatorX Why eastern europe?
@@ToddlerAnnihilator666 From what I've seen from some online math groups, there seems to be a stereotype of certain areas and their math faculty, I couldn't elaborate too much as I don't quite understand it or have the experience to speak on it's validity
@@soupy5890 Ok you are indeed correct. I was lucky to attend uni with fairly down to earth staff/professors. But those that transfered from other unis say that to keep their employment rates (and other statistics) higher they purposely fail students after certain qouta has been met
Revenge arc. Inspirational.
15 years after graduating as a math major, i still have nightmares about those exams lol. No joke.
They're tough! Thanks for watching and commenting.
I took it 4 years ago (graduated with BS in Math in 2017) and had a nightmare last night!! I guess they'll never stop... sigh
What are you doing right now? As a profession.
I'm also pursuing bsc degree in maths.
The more practice with mock exam questions, the more your confidence grows. I frequently visit Mathematics Stackoverflow and MathOverflow, considerably the best hubs where I view tough exam-like questions and others' solutions. I learn much by brain-picking gifted mathematicians there. Don't mind me, as I am only an intermediate in mathematics.
@@pinklady7184 thank you for the advise, I will do the same now, that could have been added in the video.
The video is quite helpful. Thanks for the suggestions. Struggling with the real analysis right now. Questions are not super hard, but I'm nowhere near solving them. It's frustrating that after so many hours of hard-working, I still have a hard time writing down proper proof without oversimplifying or complicating things. Whereas my classmates are gifted, discussing recondite ideas with prof all the time.
I've definitely been there, it's part of the process. You can do it!
They taught us this course in our first year of university in 1998. It was a wakeup call for so many students!
I'm actually kind of surprised to hear that (at least some) math programs use real analysis as the first proof-based course. At my school, they used the discrete math as the "intro to proof" course. As a computer science major, it was hard enough for me to pass this class when we were writing proofs about sets; I can't imagine having to throw calculus into the mix!
Agree on repeatedly failing and trying then finally getting it. That's probably the best thing that I gotten out of being a math major. There's no way I'm giving up on anything after completing my degree.
The secret to life baby!!
I totally feel you. The key here is really not giving up. I am taking analysis right now and it’s extremely difficult to me. But I still force myself to try again and again.
An undergrad course on topology without proper motivation *is* the killer.
It definitely can be!
It's killer with motivation also😂
Killer in what way?
For context I'm considering taking Real Analysis as my elective (mainly out of curiosity and for the challenge) but I've heard often that Topology helps a lot with understanding real analysis and should be taken before analysis.
@@FsimulatorX If the presentation is just abstact and formal, then you might ask 'what is all this, and what are the practical examples that lead to these definitions?' For example, the Euclidean space R^n is a prototype for a toplogical space. A well motivated book is Topology now! by Messer".
I came here after crying my eyes out because of a real analysis assignment. I really wish I knew this beforehand, it’s so disheartening to be in a class where you feel so inadequate
It can definitely be a struggle. (especially at first) You can do it!
It's the first time I feel like I'm personally advised about going for a math major. Thank you.
Happy to help!
I’m taking intro to Real Analysis this fall. So glad I found this video. Thank you!
Best of luck!
I would love to see a top level summary of what you learn in real analysis and what it allows you to go into next.
Here's a nugget: Take Set Theory if it's offered or study it on your own. This is like the "introduction" to Real Analysis. This will get you thinking about the abstract logic when studying real numbers and knowing how to "speak the real analysis language" when it's time to discuss integration and differentiation utilizing sets because you're not going to be computing hardly anything, but writing nothing, but proofs and more proofs
My university does not offer a set theory course but it does offer an Intro to mathematical concepts/proofs course that covers logic, proofs, sets, functions, relations and number theory.
any book suggestion?
or just the naive notions of set theory, not to the level akin to filters&ultrafilters
Is Algebra (not just Linear Algebra) necessary for studying Ananlisis?
This subject really shook me in my first sem class. It takes so much time to absorb, and is just So different. Watching your video's making me realize how students first upon entering college-level math need to be given intro transition classes into each subject each semester. Like, u need some storytelling and constant context to get through that!
I took Set Theory before Real Analysis I, and it was immensely helpful
I bet it was!
Any books on proofs technique you can recommend??? Great video
@@youssephfofana9226 I had a skinny little (80 pgs or so) book called Set Theory that was used in class. Sry can't remember the author but it might've been my prof (last name Bradley). Very concise explanations on axioms, (well) ordering, proof by induction, etc
@@benno291980 plz can u be more precise about the book...i could not find it. Thanks...
@@youssephfofana9226 looked it up; it's called "Naive Set Theory" by Paul Halmos
I will be teaching Real Analysis this semester. In preparation to the semester, I was planning to find a tasteful video on the history of RA. Then instead I found this video! These things came to me naturally when I was once a student, so, I was blind to most of the issues. But by experience I know, these are exactly where my students may struggle. I will share this video with my students. They'll surely appreciate it. Thanks for the excellent video!
Best of luck to you and your student this semester!
Can ask this kind of math do have number or need to caculate anything or you just write proof
You sir talk the most sense.
Glad to hear it :) Thanks for watching!
Definitions are so important as pointed out by my maths professor, who said that good defintions are 90% of the proof of theorems.
Being an undergraduate engineering student I didn't have an acquaintance with RA. I was really good at calculus, but RA kicked my butt when I got into grad school. I needed to strengthen my pure mathematics background for what I was into. And as you mentioned, it is NOT calculus. So I had trouble with the abstract nature of it. It still bugs me after all these years! And that is why I watched your video. Thanks.
"I think 90% of a math degree is just straight not giving up"
PREACH!
Real analysis was by far the most troubling undergrad math course I've taken as a math major. Don't know why I still have the textbook I don't understand 95% of the text lol
watching this when im gonna have my finals tomorrow
Best of luck!
Finals during Pandemic..Nice
Take the easy route: major in Physics.
Nope never 💀💀💀
If I ever have to teach a course on Real Analysis, this video is going to be on the syllabus
Thanks so much!
Im in university right now as a maths major straight out of highschool and my first course is real analysis 3 weeks in right now. It has been tough.
Good luck work hard and perservere.
You can do it! Best of luck!!
You can do it! .. It IS hard to digest, but get yourself s many resources you can to understand how to deal with all the elements that make up the subject. Best wishes!
As someone attempting to self study real analysis, this is so useful! This entire video managed to put into words what I’ve felt for the past month. It’s been really hard to even do the most basic proofs, but this has given me some jump off points to speed up progress. Thank you.
1) The real Analysis course is nothing like a Calculus course. It's logistically rigorous and proof-based. It's not taking derivatives, factoring, plugging, etc.
2) Be familiar with some proof techniques, such as mathematical induction.
3) Be extremely familiar with definitions: definition matters. You need definitions to start a logistic proof that your professor like.
4) Write down the definitions you know and write down what you want to prove (the conditions and the conclusions). It can be a great way to trigger a correct proof.
5) Be familiar with logistic quantifiers, all kinds of notations.
6) Persistence is key. You have to stick to learning and trust yourself. Never giving up is important to nail this class.
It’s funny how big TH-cam is. You are clearly doing a great job, you have videos that are well made and well targeted and raking in the views. This is the first video I’ve seen of yours. Keep up the great work!
I appreciate that!
Great video. Here's what you must know before becoming a math major. Real Analysis (aka 'Advanced Calculus') is usually the first proof-based course in college and it isn't normally taken until FAR TOO LATE to realize you don't understand proofs at all and you may NEVER be able to handle proof-based courses, making a math degree effectively unattainable for you. Real Analysis is usually taken in your junior year and these proof-based courses make up your last two years of college yet they are a COMPLETE CHANGE OF DIRECTION from what you thought math was all about. It's almost criminal that this isn't revealed and emphasized EARLY ON in college, but it is not. Math profs at university have pure math PhDs and were born knowing how to do this stuff (no kidding). They assume you are the same. College advising sucks and no one will tell you the following. Being good at 'plug 'n chug' will not help you with proofs. You can be an A+ student in all calculus courses and fail proof-based courses such as Real Analysis. If you happen to be reading this before it's too late, get Laura Alcock's book 'How to Think About Analysis' and STUDY IT CAREFULLY. If you can't follow her material, you are probably in the wrong major. Her presentation is, by far, the best I've seen. Even applied math majors will normally get a big dose of proof-based courses that may stop them - dead in their tracks. Get lots of help or change majors, NOW! Asking the prof for help is a waste of time. Get online and find someone on Wyzant or something like that who is good at pure math and do one-on-one tutoring. But, good luck! If you can't do proofs, a Math PhD is not going to happen. So, even if you complete an Applied Math program, you better have computer science skills in order to get a job in a relevant field (except teaching public school). You won't be teaching at University without a PhD in math.
Stoooop u r gonna make me cry 😭😖😫 I’m taking it rn!!!!
@@xHannaHx33 How is it going?
YamahaC7SRG it is the 2nd week and I’m lost 😭 can’t find good TH-cam profs that explain it well!!!
@@xHannaHx33 What book are you using and what section are you lost on? If I can find the book online, I'll take a look at the section. Remember, get help ASAP. Don't wait. Perhaps try Wyzant. Upper level math is not commonly found on TH-cam. If this is your first 'proofs' class, it will make or break you.
YamahaC7SRG AFriendlyIntroduction toAnalysis (2nd edition). It’s my first proofs class and it’s online which honestly isn’t helping. What’s wyzant??
oh, I started real analysis book by myself just bc I found it interresting to learn something abstract myself PS: i am electrical engineer at high school. Nearly every question seems unsolvable while deffinitions are so simple and logic. I really felt stupid bc I couldnt proof such clear things like intersection with such simple deffinitions. But now I see that it's not that simpel as I though))))
I hated that I didn't know proof as well as I needed, so I took a break, studied a HUGE amount of proof-related material. It's an advantage you have as an older student where you don't care how long it takes to graduate. My goal was simply to not complete a course/class unless I considered myself to have mastered the material.
Yes, yes, his point number 4! Sometimes, just start "Since, (definition) and (definition), then because of .... we see that (conclusion.)" Write this in "math" to practice. Trust me, you will be on your way.
I'm a freshman in an introductory proof course, and I have a feeling that this will really help me out later on
I hope it does. Best of luck!
What's book do you use for introduction proof???
The name and the author precisely the ISBN# will be great. I'm looking forward as independent study.
Thanks...
@@youssephfofana9226 hmm I don't have the ISBN, but it's called "Analysis with an Introduction to Proof" by Steven Lay. Here is the link: www.pearson.com/store/p/analysis-with-an-introduction-to-proof/P100001370585/9780321747471
I come back to this every time the textbook feels like its killing me
Knowing, not just understanding, but knowing definitions verbatim, agree 100%. It was the thing I had the hardest time finally understanding.
2:38 induction
definitions. alarm bells.
I remember when I was sitting in an upper-division class and remember the panic when the professor would right Defn. -
Now I understand. LOL
5:20 Start at what You know and The answer you want, meet the 2.
6:30 Notations, quantifiers, symbols,
Currently doing a major in applied math and computer science and Real Analysis was one of the most dreaded courses. Unfortunately got stuck with Real Analysis, Complex Analysis and Stochastic Processes all in one semester haha. But who doesn't love a good challenge ;)
lol how did it go man? I'm thinking of taking real analysis, abstract algebra and stochastic processes on the same semester. Complex Analysis sounds dreadfully worse though XD
I did Real Analysis some 25-26 years ago & I still get nightmares about it every now & then. But I’ve learned to appreciate it more over the years.
Real Analysis as the FIRST proofs-based course? That's insane, I had my own introduction to proofs class and apparently talking to some computer science students my introductory class was three discrete math classes combined. Then I picked up a Real Analysis book to see what it's like and the first Unit just casually summarized everything I learned in that class. I'm still trying to read the book; it's intense and the exercises are brutal, but I find RA VERY rewarding. I was forced to switch Real Analysis with Abstract Algebra and Probability Proofs for the upcoming year, but hopefully I can take RA soon.
I agree with you about not giving up and persisting. Sometimes the first encounter with some new material is like hitting an impenetrable wall. You have to persist in battering that wall, until it gives way to your understanding.
Not giving up is the key! Can't tell you how many times I had to repeatedly stare at problems or theorems before I could get my head around it.
@@BriTheMathGuy When I first tried abstract algebra I saw the material and just about had a panic attack. I am not ready for it yet. I am brushing up on my fundamentals first before I try again.
All of this is why they should teach Real Analysis *before* Calculus.
If you're balancing a checkbook, you don't need Calculus.
However, if you're using Calculus, you need to understand the tools you're using.
so glad i was able to take geometry and abstract algebra before RA. i had never seen non-E geo before then, so that solidified the importance of definitions. the set theory and structural concepts of ab. alg. helped me get my mind right. RA was still a challenge, but it’s like i was looking at calculus from “the other side,” so i really felt like i was going somewhere in my math understanding
I watched this video before starting the real analysis course so I couldn't understand what he was talking about back then. Now, 2 months later, I can understand and relate to each and everything he is saying. I followed what he told and I can say I am starting to get a grip, still very very very far from the end but I am moving atleast.
Edit: I think I'm gonna fail this semester 😭
Update???
My God, what you just said here about the essential importance of knowing definitions really opens my eyes to so much, not just in math, but in life and the world in general...
Thank you. This is a very interesting and useful video. The best of this sort I've ever seen.
Glad you enjoyed it!
Issue:
What is a differential of an irrational argument?
Let a= some rational approximation, and A be the irrational number itself (if that makes sense).
Then A - a > dA and there is no way a + dA > A
Re. Real Analysis vs Calculus sequence:
In some teaching sequences, Real Analysis is taught _before_ Calculus. There is a certain logic in doing so, theory before application. Hence why Calculus was usually feared being the final class taken. In the most of US (United States) for the last 40 (or 50?) years, that is reversed because more practical teaching that way.
In Mathematics Departments' defense, they are teaching to a much larger audience than potential math majors; Chemistry, Physics, pre-med, econ, engineering, Comp Sci, and possibly any other are all required having mathematics, and in most cases, Calculus and Differential Equations. So makes much more sense that the first year(s) courses are calculus then switch to the more rigorous theoretical framework of upper division (junior & senior level) classes which are geared specific toward math majors. Unless one is doing Harvard's Math 55, they are not likely to study Real Analysis in their first year.
I am watching this video half an hour before my Real Analysis exam. I hope it’s not too late
Definitions are the times tables of higher math. Know them cold and they will serve you well.
It annoyed me into oblivion that in my math class I had to do "proofs" that weren't proofs - more like a basic vague sketch of ideas that really has no structure - at all and it was almost like the teachers were actively trying to persuade us that proofs don't exist. I can't even talk to my classmates now, because just a slight derail from the beaten path of plug and chugging leaves them confused, and that's kinda sad, being unable to really share these great ideas with anyone as an EECS undergrad(At least I hope that's the reason) :D It's a lot of work to learn this properly though
This is so true! I'm in my final year as an Electronics undergrad and in most of our classes that involve maths to some extent, the teachers really don't care about rigour and we just take most things for granted...
Having accomplished 2 Math Degrees so far...I highly recommend a std Web Search for:
'real analysis prerequisite courses', reason being that there are a couple of basic lower-level Math courses which WILL help greatly such as:
*Discrete Math (emphasis on Logic & Induction, Boolean Algebra, etc)
*Abstract Algebra I/II (sometimes called Mathematical Methods in undergrad programmes)
*Introductory Analysis (basic methods of proof, etc).
However, apart from these there are the more 'specialized' courses route such as:
*Advanced Linear Algebra
*Set Theory
*Point-Set Topology
In hindsight, I presonally summarize Real Analysis as (Advanced) Calculus I-III using R.A. methods
& similarly for my personal Major & all-time favorite topic 'Complex Analysis,' Calculus I-IV using C.A. methods.
Footnote: make no mistake, these advanced 'Analysis' type courses require committment, dedication & HARD WORK to SUCCEED! Be prepared to FAIL (& YES I failed, did a few repeats along the way), but KEEP practising as commented in this Video. You'll eventually appreciate the sheer beauty, mystery & wonder of the mathematical Universe...as I have. So, good luck on your journeys, my mathematical friends!!
Interesting.
I see how it would be difficult to unlearn Calculus from the geometric approach as it is introduced and developed in the usual way, then relearn it from an arithmetic/algbraic/logic point of view.
*Me:* "I understand Calculus."
*ANALYSIS:* "You don't understand the Number Line."
One small thing I would like to add too is the use of counterexamples. I didn't fully understand this at the time when I took Real. The idea that for some proofs (not all) you can use a counter example that contradicts the claim. It's easy to overthink this as it's hard to think of an example that contradicts the claim.
One of the most difficult thing for me as a beginner is to use mathematical definitions to proof certain theorems...these theorems makes you realize that you don't know the actual meaning (essence) of the topic.. 😑😑
I couldn't do it in university - it was my 8th class, I had 7 other courses. The pressure was too much so I dropped it and the idea of getting a double major in math and engineering. Without the pressure I now use it to escape the insanity around me and find it very interesting - lots of little puzzles but no harder than puzzles you would find in a book of puzzles. Of course, a book of puzzles is a waste of time since you can get the same pleasure and learn something practical and useful like real analysis. If you take your time, it's not that hard - stop at something you don't understand until you do - if that takes a week, no big deal. I don't know why anyone would solve puzzles when a desire to be challenged can be met with so many useful subjects like philosophy and math and these really develop your mind. You know it's working when you start to find sitcoms and movies boring. I'm 56 and can still learn this kind of stuff although it is harder than when I was in my twenties.
It means that for every epsilon greater than 0, there exists an n in the Natural Numbers such that for all n greater than some k (also a natural number), then the absolute value of the sequence (X_n) is less than epsilon. There!
If only I discovered this video before going in... things would have been very different. All those things are so true..
Glad you thought so. Thanks for watching and have a nice day!
I had a master of science degree in mechanical engineering and even had taken two graduate course in Classical Mechanics and Mathematical Physics when I decided I wanted to get a masters degree in mathematics. Real Analysis was the first course. I was blown away and had to drop it. I wish this video had been around all those years ago! It was like approaching math in a completely different way. I just couldn't wrap my head around it.
The key tips are really amazing and helpful.I hope to see more of this
Real Analysis was difficult for me as an undergraduate at UCSC…fact is, I didn’t pass the first time.
The next year I met Ralph Abraham who was teaching at UCSC at the time (ca. 1982).
He said that there were two ways to understand mathematics….
One was the brute force method in which you struggle for hours on end to understand the problem.
He said that this method will be difficult, but can certainly succeed.
The other, is to think about the problem before you go to sleep at night… If upon awakening the next morning, you have the answer to the problem
then you will know that the path of mathematics is the right path for you. I asked why this was…he said that much of mathematics lies in the dream world. For example, the idea of complex numbers only makes sense in the dream-world. He said that this was the only way to truly understand Mathematics.
And it does work…I could not have received a degree in Mathematics otherwise.
RIP my friend.
Do you have any tips on how to (improve) remember(ing) math theorems and definitions? Most of the times I sort of get the point of how you can use it and understand what it does, yet I always struggle to memorize the more precise/small parts like for example some given requirements, preconditions or exceptions.
Good question.
I think it's achieved by solving problems
You gotta understand certain concepts at a very fundamental level
I am in Real Analysis now :) And I am elated with the concept of convergence and the Topology as well! I am fond of Topological Groups!
This channel is Gold.
Thanks very much!
Thanks Bri! What you're providing are the axioms to have when you walk through the classroom door. That way, Real Analysis axioms are the second set of assumptions you learn instead of the first!
Had to take this is a Computer Science Major 😭😭😭
I thought Maths was easy before real analysis. I still have nightmares
Thank you so much!! Starting Real Analysis later today.
Best of luck!
But 50% of the screen was literally unused the whole video 😂
Is that supposed to be some metaphor!
I studied Measure Theory without any knowledge about Real Analysis. For 3 months up until now I studied every day, starting from the definitions of open sets, etc. I still remember the first several classes I could not understand what prof said. So I just quitted the class and studied it myself. Now I can understand the proofs and the solutions of exercises, but I feel like I’m still far away from solving exercises 😂. Tomorrow I have exam. Wish myself luck 😅
Did you pass
@@gauravbharwan6377 Yes! 16/20. I came back right after I had result. Turn out one (over 3) exam question was similar to one exercise and I learnt it by heart, so... well
@@NgocAnhNguyen-si5rq I am also facing bad college teachers
Taking PSU MATH312H (Intro Real Analysis) right now so the timing of this video was perfect
Good luck.
Great content. Thanks for sharing your insight
Thanks very much! Have a great day.
thank you for spending your time to create this video.
Taking it in the first semester as a Physics major, wish me luck!
Goddamn... good luck!
How was it?
Real analysis is about the core nature of real numbers.I personally feel the reason why students find it difficult is that they study functions,limits ,differentiation ,integration and sequences intuitively.
Most students do not have a sound knowledge of set theory and logic and therefore lack rigor in their veins.
Really cool, interesting advice, thank you!
Real analysis takes time to get it .Don t panic!do ur HW.most professors gives 90% exams similar like HW.stay positive and you can do it with A+.
Agree with this completely. I was just thrown into real analysis (it was my very first lecture in university) and had to painfully realise this myself as the lecturer didn't say it.
As an ambitious high schooler looking to major in mathematics and have given a few attempts at real analysis, this was incredibly helpful
Check our Terence Tao's book, the first one is super readable. Way better than Rudin's analysis. After Tao go to Rudin and it'll be easier.
Gustavo López much appreciated! I just started using Bartle and Sherbert's Introduction to Real Analysis or something. What's the title of Terry's book?
@@marks4982 Analysis 1 and Analysis 2. However, if you are in high school I recommend " A story of real analysis". It covers a typical first course in analysis with a lens of history describing how the subject evolved and providing motivations for dry definitions. The mathematical maturity you'll develop after completing this book will be great. I also used this in high school. You need to know differential and integral calculus ( because it treats examples that way) and some basic differential equations.
I read and went through all the questions from Richard hammock's book of proof. I don't know if this is the same this as pure logic, but it is beautiful. To me, real analysis sounds like it builds on the bones of logic. But what kind of course or subject lies at a deeper more fundamental level. Like the three rules of thought and a proof for the existence of truth.
Physicist here; I took Real Analysis in grad school and loved it.
I studied real analysis with the late Prof. Rudolf Vyborny, a Czech emigre, in the 1980_s . It was one of the most intellectually fulfilling studies in my life.; but it is hard and challenging for sure. I would encourage anyone who feels a bit overwhelmed by the rigour of the subject to persist and to believe that they will get there in the end.
Fabulous expression and ideas. Especially the 3rd of them which can make me more clear in proofing, I guess ;D. Thanks a lot~
Velleman's How to Prove It opened the world of math to me. I thought I'd need proofs for Calc 1, but aside from delta/epsilon🤷♂️ Everyone should be obliged to take a course based on Velleman before embarking on a math major. He goes thru symbolic logic, quantifiers, everything.
I took Real Analysis in the early 80's and I wish I had this video to watch back then before I took it. I honestly was not ready for it and the prof (who wrote the textbook) never really gave any overview on the topic or discussed any of the techniques required to do the proofs. He kind of assumed we knew how to do "Proof by Contradiction" or "Proof by Induction" etc. In fact, one of our assignments was to prove the Dedekind Cut Theorem. I think everyone got lost on that one.
Lol watching this one day after my real analysis exam has me like, wow....CAN DEFINITELY RELATE!!!
By the way, you're the one who taught me what a topology is. Absolutely love your video on that, it was extremely helpful and I love this video.
It's ℝeal lol.
It's sorta sad the way the math program is made though, before this level, no one will know what's about to hit them. I only knew that everyone said they "hate" real analysis. Never heard nor saw anything else about it. Now I understand that it's an analysis based course, which I kinda like tbh, and kinda hate.
I loveeeeee finding solutions to problems lol it makes me feel like a jackpot winner LOL, but on the other hand, I HATEEEEE all of these seemingly random, yet connected and absolutely necessary theorems and definitions.
I stopped counting them when we hit topology. LOL.
I do however have a very LONGGGGGG list of almost all of the definitions and theorems stockpiled into a MS Word document hahaha...
Crazy stuff.
Tbh, Advanced Calculus was much harder for me than ℝeal analysis because at least by the time I hit ℝeal analysis, I was already exposed to a great many of the definitions that we would need to use.
I’m taking real analysis this next year, and I feel relatively good with the things you mentioned since I’ve already taken an intro proofs course, and 2 abstract algebra courses. The only thing that worries me is the fact that I haven’t taken any type of calculus in 2 years. But nevertheless I still feel confident.
you don't really need to 'know' calculus, but it does help knowing calculus so that the motivation behind learning analysis is there. When you know the end goal, you gain a greater appreciation for the machinery they build up in the class.
So how'd it go?
Thank you for this pep talk
If people knew what lie in store for them, they would pay more attention in calculus to the proofs of the theorems, not just the applications. I remember I had the "I just want to know how to do this" mentality. As Brian mentions in his videos, you are in for a rude awakening if you do this once you get to real analysis.
About the sandwiching "technique" that is mentioned here: I remember using it in Euclidean Geometry in my early teens. Then I considered it natural so that everyone should be using it. I tried to teach it to one student two years my junior, but it seemed easier said than practiced for him.
Thank You for your suggestions ✌
Proofs class first. Definitions, yes!
This was so helpful.
In my current undergrad experience its interesting the courses which definetly are built upon complicated niche rigorous proofs are often the classes where you arent required to do those proofs, because many students who find calculus appealing end up reaching into areas like differential geometry and geometric calculus those courses are often taught at the undergrad levels with out too many proofs even tho there “past” calculus they require real analysis to work
This Video Is Very Helpful For Those Students Who Are Going to Study University level Mathematics
Great video, I graduated from my math BA program in January and many concepts that were still fuzzy are finally starting to click now and it happens at the most random times so I would agree that persistence is key, I don’t think I would have done well in Real Analysis if I had not taken a proof course first. Set theory is your friend 🙂
Awesome video! Thank you!
Glad you liked it!
Before taking the class, all this advice sounded obvious. After taking the class however, all this advice sounds painfully personal. Suffice to say i had to go over all the proof-based classes again after finishing the first year.
In days past, there wasn't a single "proof" class offered or available. You took the 3 core Calc classes, an ODE & Linear Alg. class then, BOOM you're were in a Real Analysis using baby Rudin's book. Professors assumed you knew how to prove things and it was sink or swim. There were very few students that made it. I guess the Math Dept. used this as one of their "weeding out" courses.
The divide between school level calculus and undergraduate calculus/analysis applies to pretty much all subjects of mathematics.
You think arithmetic is easy? Wait till you learn number theory.
You think polynomials and theory of equations are easy? Wait till you learn abstract algebra.
You think linear algebra is easy? Wait till you learn linear algebra, (but done right).
And of course finally:
You thought baby Rudin was hard. Just wait till you read papa Rudin.