Proof: Sequence Squeeze Theorem | Real Analysis 

Thanks a lot, glad you find the explanations useful!

You're very welcome! Glad it was clear!

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!

Thanks Mallyn, that's always what I am trying to create! If you haven't already, check out my analysis playlist for more! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

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

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! ru-vid.com/group/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli

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

Thanks for watching! Let me know if you ever have any questions!

@@WrathofMath my book says this: Proof. Let ε>0. Choose k1 ∈N so that if n≥k1 then |xn−c|

Glad to help!

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.

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!

Thank you!

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

Where do you teach?@@WrathofMath