Real Analysis Convergence Question

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Homework Help Overview

The discussion revolves around proving limits in real analysis, specifically focusing on the convergence of sequences and the application of mathematical induction. The original poster presents two problems: one involving the limit of a sequence defined by \( (1+k/n)^n \) and another concerning the limit of the difference between two sequences given a specific condition.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • The original poster attempts to use mathematical induction for the first problem and questions whether the same process applies for general \( k \). For the second problem, they express uncertainty about the next steps after establishing the boundedness of \( y_n \).

Discussion Status

Some participants provide guidance on the induction approach, suggesting that proving the case for \( k+1 \) follows from the case for \( k \). Others discuss the second problem, noting the importance of the definition of limits and suggesting a strategy involving the epsilon-delta definition. There is a mix of supportive and critical feedback regarding the methods proposed.

Contextual Notes

Participants note the boundedness of the sequence \( y_n \) and the implications of the limit condition \( \lim x_n/y_n = 1 \). There is also mention of etiquette in the discussion, reflecting on the tone of responses and the appropriateness of language used in critiques.

Askhwhelp
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1) Use mathematical induction to prove that for any k ∈ N, lim (1+k/n)^n = e^k.

I already used monotone Convergence Thm to prove k=1 case. Do I just need to go through the same process to show k? If not, could you please help?


2) Suppose that ( x_n ) is a sequence of real numbers, ( y_n ) is a bounded sequence of non-zero real numbers, and that lim x_n/ y_n = 1. Prove that lim x_n - y_n = 0.

Since y_n is bounded, there exist M such that |y_n| <= M for all n in N. Then what should I do?

Thanks
 
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1) Next you show that if it is true for some k, then it is true for k+1. You are anchored on k=1, so the glide
from k -> k+1 takes care of the rest.
 
Askhwhelp said:
2) Suppose that ( x_n ) is a sequence of real numbers, ( y_n ) is a bounded sequence of non-zero real numbers, and that lim x_n/ y_n = 1. Prove that lim x_n - y_n = 0.
2) To solve the problem you have to show for every ε>0 there exists an N .st. if n>N then |x_n - y_n|<ε. Now since x_n/y_n-->1 , given an ε>0 there exist N .st. if n>N then 1-ε< x_n/y_n <1 +ε. And now given that y_n>0 makes the next step easier. (When proving limits always go back to the basic definition to see where you need to go). Anyway now you are in business...
 
Last edited:
UltrafastPED said:
1) Next you show that if it is true for some k, then it is true for k+1. You are anchored on k=1, so the glide
from k -> k+1 takes care of the rest.

BTP deleted the response to post 1) but I'll echo it. It's insane to do this by induction if you are anchored on k=1. I don't even see how you would do it. It's just a change of variables. 1+k/n=1+1/(n/k). Change the limiting variable to n'=n/k.
 
I wasn't sure of etiquette so I pulled my insane comment. But now I know.
 
BTP said:
I wasn't sure of etiquette so I pulled my insane comment. But now I know.

Calling a person insane is one thing. Calling a question strategy insane is another.
 
Dick said:
Calling a person insane is one thing. Calling a question strategy insane is another.

Ha, I got the not calling a person insane part. I wasn't sure about calling a problem insane. Cheers!
 

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