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Real Analysis : Convergent and Cauchy Sequences - Five Problems

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? For each of the follwing statements decide if it is true or false. Justify your answer by proving, or finding a couter-example.

1) every bounded sequence of real numbers is convergent.
2) Every convergent sequence is monotone.
3) Every monotone and bounded sequence of real numbers is a cauchy sequence.

? For each of the following statements decide if it is true or false, justify your answer by proving or disproving the statement:

1) a sequence of real numbers can have different subsequences which converge to different limits.
2) A convergent sequence of rational numbers is a cauchy sequence.

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Convergent and Cauchy Sequences are investigated. The solution is detailed and well presented.

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• For each of the follwing statements decide if it is true or false. Justify your answer by proving, or finding a couter-example.

1) every bounded sequence of real numbers is convergent.
False, a counterexample is :
1,-1,1,-1,1,-1,....,(-1)^(n-1),...

2) Every convergent sequence is monotone.
False, a counterexample is

It is convergent with limit 0 but ...

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."
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