a) False b) Follow from Theorem 5 c) False
Work Step by Step
a) This statement is false. The sequence $a_n=cos(\pi n)$ is bounded since $-1 < cos \pi n \leq 1$ for all $n$, but it does not converge: since $a_n=cos(\pi n)=(-1)^n$, the terms assume the two values 1 and -1 alternatively, hence they do not approach one value. b) By Theorem 5, a converging sequence must be bounded. Therefore, if a sequence is not bounded, it certainly does not converge. c) The statement is false. The sequence $a_n=(-1)^n$ is bounded, but it does not approach one limit.