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