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Real Analysis : Limits and Cluster Points

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Please see the attached file for the fully formatted problems.

1) Prove that does not exist but that .

2) Let f, g be defined on to , and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that .
Prove that .

3) Let f, g be defined on A to and let c be a cluster point of A.

(a) Show that if both and exist, then exists.
(b) If and exist, does it follow that exists?

4) Let f : be such that f(x+y)=f(x)+f(y) for all x, y in . Assume that exists. Prove that L = 0, and then prove that f has a limit at every point . (Hint: First note that f(2x) = f(x)+f(x) = 2f(x) for . Also note that f(x) = f(x - c) + f(c) for x, c in )
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Limits and Cluster Points are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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1) Prove that does not exist but that .
I think the statement should be " Prove that does not exist".
Proof. Let and be two sequences. It is easy to see that

So,
So, does not exist

2) Let f, g be defined on to , and let c be a cluster point of A. Suppose that ...

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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