Start Learning Reals 2 | Completeness Axiom
ฝัง
- เผยแพร่เมื่อ 9 ก.พ. 2025
- 📝 Find more here: tbsom.de/s/slr
👍 Support the channel on Steady: steadyhq.com/e...
Other possibilities here: tbsom.de/sp
You can also support me via PayPal: paypal.me/brig...
Or via Ko-fi: ko-fi.com/theb...
Or via Patreon: / bsom
Or via other methods: thebrightsideo...
Join this channel on TH-cam: / @brightsideofmaths
💬 Access to the community forum: thebrightsideo...
🕚 Early access for videos: thebrightsideo...
❓ FAQ: thebrightsideo...
🛠️ What tools do you use: thebrightsideo...
Please consider to support me if this video was helpful such that I can continue to produce them :)
Each supporter gets access to the additional material. If you need more information, just send me an email: tbsom.de/s/mail
Watch the whole video series about Start Learning Mathematics and download PDF versions, quizzes and exercises: tbsom.de/s/slm
Supporting me via Steady is the best option for me and you. Please consider choosing a supporter package here: tbsom.de/s/sub...
🌙 There is also a dark mode version of this video: • Start Learning Reals 2...
🔆 There is also a bright mode version of this video: • Start Learning Reals 2...
🔆 To find the TH-cam-Playlist, click here for the bright version:
🌙 And click here for the dark version of the playlist:
🙏 Thanks to all supporters! They are mentioned in the credits of the video :)
This is my video series about Start Learning Sets. We discuss basic definition, like union and intersection, and also go into the details of maps and functions. I hope that it will help everyone who wants to learn about it.
x
#StartLearningMathematics
#Mathematics
#LearnMath
#calculus
#realnumbers
#numbers
x
I hope that this helps students, pupils and others. Have fun!
(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)
For questions, you can contact me: steadyhq.com/e...
The Bright Side of Mathematics has whole video courses about different topics and you can find them here tbsom.de/s/start
Good stuff as always! Looking forward to the construction of the reals! :)
loving this playlist, beautiful.
i can follow the story of the reals.. thanks so much for this.. hoping for more
At 3:30 in the video, It is stated that a is an element of the rational Numbers, And that there is a sequence Xn that converges to some number, such that There always exists an 'a' where |Xn-a|
Limits have to lie in the space you consider, otherwise the sequence would not be convergent. The sequence you want to consider: 1, 1.4, 1.41, 1.414,... is not convergent in the rational numbers. It has no limit.
However, in the real numbers the same sequence has a limit.
Thank you so much for the video!!!!
Very good work sir .
Nice explaination
For some reason whenever I go back reviewing real analysis (e.g. in preparation for functional analysis), I feel like I have "quantifier dyslexia" sometimes: e.g. if I haven't seen the definition of a cauchy sequence, even though I read the quantifiers in order I will mix them up (and I think that turns it into a statement that can't be satisfied?), and then I go back putting it into words again like you are doing, and Tao also happens to do in his book where he describes "eventual epsilon steadiness" as:
- 'eventual epsilon-steadiness' means there exists an Nth step in the sequence where it becomes 'epsilon-steady', (∃N ∈ ℕ ∀ n,m ≥ N : |xₙ − xₘ| < ε)
- and 'epsilon-steadiness' means every pair of sequence members x_j and x_k are separated no more than epsilon apart, (∀ n,m ∈ ℕ : |xₙ − xₘ| < ε)
- i.e. their distance is contained in the epsilon neighborhood,
- or i.e. each every pair of sequence members is 'epsilon-close' (|xₙ − xₘ| < ε)
So putting it all together: there exists an Nth step in the sequence such that every pair of members x_j and x_k after the Nth one are epsilon-close
Whereas the statement with the first two quantifiers reversed (for cauchy sequence) would appear to be: "there exists an Nth step in the sequence such that you can shrink the epsilon-neighborhood arbitrarily close and all sequence members after the Nth one will be contained in it"... seems like that can only be satisfied if all those sequence members after the Nth one are the limit point itself (assuming we're in R).
7:24 in this proof is a uniquely defined? as in if any other numbery say b which also satisfies the same steps does that imply b=a? and how would one prove this if it's true?
ohh nvm a comes from the convergence implication and is not a random number lol
Wonderfull
Thank you! Cheers! :)
Thank you so much for such videos, 1:10 Is triangular inequality and inverse triangular inequality explained later ?
Is the arrow at 4:31 a conditional operator? And if yes, I would understand it as an operator on the statements A: the sequence is Cauchy and B: the sequence is convergent. However, that would mean that even if B was true and A was false, the statement given by the operator would be true since the double arrow gives a tautology. That would mean that if a sequence was convergent it does not have to be Cauchy? Or am I missing something here :) ?
Very good question: I introduced the notation you mentioned in "Start Learning Logic - Part 3". However, we already reached a point where one usually does not use a strict formal language anymore like we did at the beginning of logic. In other words, the correct way to write it here would be: "(B → A) is true".
The problem is, the symbol (→) with all the maps, functions and so on, at this point, is totally overstrained. While creating this video series, I really thought a long time about this conflict. In the end, I decided that it is better to leave the normal conditional (→) behind.
@@brightsideofmaths Thanks for the answer! Indeed I am referring to the video "Start Learning Logic - Part 3" ;) Have a nice day :)
Very didactic!
7:51 what does it mean by ordering less than or equal to?
It's an order relation. So it satisfies the rules for it.
@@brightsideofmaths Thanks for responding, by the way, I have one last question. When proving that the Convergent sequence is the Cauchy sequence, around the 6:54 mark, how exactly does this show that they are the same?
I guess I'm just a bit confused on what it means by being the same. I'll type here what I know:
We started the proof by having the distance between two sequence members |xn-xm|. We know by absolute value that,
|xn - a+ a-xm| = N, we have |xn - a| < epislon prime.
This leads us to seeing that, since both |xn - a| and |a - xm| are less than epsilon prime, their addition will be less than epsilon. So,
|xn - xm| = |xn - a + a - xm|
I am not sure I understood your confusion but they are not the same. What was shown was that a convergent sequence is definitely a cauchy sequence, not necessarily the other way around.
Hi !
At 7:15 , how did we know that " |a-xm| < Ɛ " ?
The missing step is to prove |x| = |-x| for all x, and then we get |a-xm| = |xm-a| which is less than epsilon because m≥N
@@AssemblyWizard thanks
Sir, I see some other definition of completeness axiom being that "for every set A that is a subset of S, supremum of A always exists". How is this related to the "cauchy sequence being convergent" definition?
Yes, this is what we discuss when we talk about supremum.
I am not sure why the triangle inequality holds, I can't think of an example where the LHS is not equal to the RHS
Some negative numbers? :)
gotta do bit proof
Do i need to study something else as a base before studying this?
I mean i just finished high-school but i want to learn more, so i dont know if i need a base knowledge for this
@@Anonymous-rj2lk You can watch the series here
th-cam.com/play/PLBh2i93oe2qtbygdXz4u6Mkh7c_hMLBA8.html
You need some mathematical thinking but not much knowledge. Just try and tell me if it worked :)
@@brightsideofmaths its just that i dont understand the mathematical writings, i do however understand the big idea but it gets a lil confusing with the unfamiliar symbols
@@Anonymous-rj2lk Then maybe my videos can help a little bit!
@@brightsideofmaths much appreciated! Thank you!