Proof: Sequence Squeeze Theorem | Real Analysis
ฝัง
- เผยแพร่เมื่อ 25 ส.ค. 2024
- We prove the sequence squeeze theorem in today's real analysis lesson. This handy theorem is a breeze to prove! All we need is our useful equivalence of absolute value inequalities that we use all the time, and to remember the definition of the limit of a sequence, and we'll be good to go!
#RealAnalysis
Definition of the Limit of a Sequence: • Definition of the Limi...
A Useful Absolute Value Inequality: • Proof: A Useful Absolu...
Real Analysis Playlist: • Real Analysis
★DONATE★
◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: / wrathofmathlessons
◆ Donate on PayPal: www.paypal.me/...
Thanks to Robert Rennie and Barbara Sharrock for their generous support on Patreon!
Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: crayonangel.ba...
Follow Wrath of Math on...
● Instagram: / wrathofmathedu
● Facebook: / wrathofmath
● Twitter: / wrathofmathedu
My Music Channel: / @emery3050
You have a great way of breaking down and explaining proofs. Thanks!
Thanks a lot, glad you find the explanations useful!
You explain the pinching theorem of sequences in a clear way. Thank you so much! I will certainly review your proof on a regular basis.
You're very welcome! Glad it was clear!
Me and my girlfriend love your videos - you've really helped her with your graph theory videos! Please keep it up - you are criminally underrated.
Thanks so much, Richard! Glad they've been helpful and I hope you'll both continue to find the upcoming videos valuable! Lots more Real Analysis and Graph Theory to come this year!
this is probs the best explanation for this proof, thank you so much!
Thanks Mallyn, that's always what I am trying to create! If you haven't already, check out my analysis playlist for more! th-cam.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Really Glad I found your page! Thank you.
Beautiful proof.
It is indeed! An intuitive result with an equally smooth proof!
You are a life saver
Glad to help!
Thanks very much!
can u please explain in a more simple way ..why the new N we took to be the maximum of N1 & N2 ....i understood the rest of the part as u explained it veryyy nicely and thank you soooo much for it :)
Thanks for watching and for the question! This is a strategy we'll see a lot in various analysis proofs. To show x_n is converging to L, we need to get a_n and b_n sufficiently close to L. We know a_n is sufficiently close to L for all n after N_1, and we know b_n if sufficiently close to L for all n after N_2.
Then, I need some other N value, after which x_n is also really close to L. Suppose I take N to be N_1. Then I know a_n is sufficiently close to L for all n past N, since N is N_1. But I don't know that b_n will be close to L for all n past N, because I don't know how N compares to N_2. If N is smaller than N_2, b_n might have some terms after the Nth term that are still too far from L. I need both a_n and b_n to be really close to L because I need to use that info to show x_n, which is squeezed between them, is also really close to L. So, to ensure both a_n and b_n are sufficiently close to L, I can't just take N = N_1 or N=N_2, I need to make sure N is at least as big as both N_1 and N_2, and so I set it to the maximum of the two.
It's harder to explain this stuff over text, so that might not have helped, but hopefully it did a little! Check out my analysis playlist for more! th-cam.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
@@WrathofMath Thank you for answering🌟..and Yes I got an idea about it..
I hope you get more visibility!
Thank you! If you know anyone who would benefit from the analysis lessons, please share them and let me know if you have any questions or video requests!
But set brackets and parenthesis denote different things:
(an)=0,1,0,1,0,1,...
{an}=0,1
Are they used interchangeably?
Different texts use different notations for sequences. I was originally most familiar with the { a_n } notation and so used that for some of the earlier real analysis videos. But I have taken to the ( a_n ) notation which is what I generally use now.
beautiful
Thank you!
Are you British or American ?
Edit : forgot my main comment 😅. Love your videos. I will ask my dad to donate and support your channel
Thanks so much - I am American! Let me know if you ever have any questions!
Where do you teach?@@WrathofMath