Real Analysis Convergence Question

In summary, the first conversation is about using mathematical induction to prove that for any k ∈ N, lim (1+k/n)^n = e^k, while the second conversation discusses how to prove that lim x_n - y_n = 0 given certain conditions. The second conversation suggests using the basic definition of limits and changing variables to solve the problem.
  • #1
Askhwhelp
86
0
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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  • #2
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.
 
  • #3
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:
  • #4
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.
 
  • #5
I wasn't sure of etiquette so I pulled my insane comment. But now I know.
 
  • #6
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.
 
  • #7
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!
 

1. What is convergence in real analysis?

Convergence in real analysis refers to the idea that a sequence of numbers or functions approaches a specific value or limit as the number of terms or inputs increases.

2. How do you determine if a sequence or series converges?

In real analysis, there are various tests and criteria that can be used to determine the convergence of a sequence or series, such as the limit test, comparison test, and ratio test. These tests involve analyzing the behavior of the terms or ratios of terms in the sequence or series.

3. What is the difference between pointwise and uniform convergence?

Pointwise convergence refers to the convergence of a sequence of functions at each individual point in the domain, while uniform convergence means that the sequence of functions converges uniformly across the entire domain. In other words, uniform convergence requires the convergence to be consistent at all points, while pointwise convergence allows for the convergence to vary at different points.

4. Can a sequence have multiple limits?

No, a sequence can only have one limit. If a sequence has multiple limit points, it is considered divergent. However, there are cases where a sequence may not have a limit, such as when the terms oscillate between two or more values.

5. How is convergence related to continuity?

In real analysis, convergence is closely related to continuity. A function is continuous at a point if and only if the limit of the function at that point exists and is equal to the value of the function at that point. This means that if a sequence of functions converges, the limit function will also be continuous.

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