# Real Analysis : Convergent and Cauchy Sequences - Five Problems

See attached file for all symbols.

---

? 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.

https://brainmass.com/math/real-analysis/real-analysis-convergent-cauchy-sequences-28850

#### Solution Preview

Please see the attached file for the complete solution.

Thanks for using BrainMass.

â€¢ 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.