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