Proof: Sequence Squeeze Theorem | Real Analysis

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
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[@nmionline9003]

You have a great way of breaking down and explaining proofs. Thanks!

[@WrathofMath]

Thanks a lot, glad you find the explanations useful!

[@jingyiwang5113]

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.

[@WrathofMath]

You're very welcome! Glad it was clear!

[@Miryr]

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.

[@WrathofMath]

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!

[@MallynMendez]

this is probs the best explanation for this proof, thank you so much!

[@WrathofMath]

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

[@zahinhuq7431]

Really Glad I found your page! Thank you.

[@Koj4]

Beautiful proof.

[@WrathofMath]

It is indeed! An intuitive result with an equally smooth proof!

[@sarahkhan7875]

You are a life saver

[@WrathofMath]

Glad to help!

[@valeriereid2337]

Thanks very much!

[@bhumicaprakash7707]

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 :)

[@WrathofMath]

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

[@bhumicaprakash7707]

@@WrathofMath Thank you for answering🌟..and Yes I got an idea about it..

[@zahinhuq7431]

I hope you get more visibility!

[@WrathofMath]

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!

[@MagnusTheUltramarine]

But set brackets and parenthesis denote different things:
(an)=0,1,0,1,0,1,...
{an}=0,1
Are they used interchangeably?

[@WrathofMath]

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.

[@animerecords7953]

beautiful

[@WrathofMath]

Thank you!

[@omjoglekar3677]

Are you British or American ?
Edit : forgot my main comment 😅. Love your videos. I will ask my dad to donate and support your channel

[@WrathofMath]

Thanks so much - I am American! Let me know if you ever have any questions!

[@AubreyForever]

Where do you teach?@@WrathofMath