Question

Write an exampale for a bounded sequence which is not convergent

Answer 1

Let us consider $a_n=(-1)^n$ . This sequence is bounded: $-1\leq a_n\leq 1$ .

Suppose that it converges: $\lim\limits_{n\rightarrow \infty}a_n=A$ .

Let $\varepsilon =1$ . Then $\exists N$ such that $\forall n>N$ : $|a_n-A|<1$ .

For $n=2N$ we have $|1-A|<1$ .

For $n=2N+1$ we have $|-1-A|=|1+A|<1$ .

And we have that $2>|1-A|+|1+A|\geq |1-A+1+A|=2$ . So, $2>2$ .

It proves that $\{a_n\}$ is not convergent.