Question

given example of convergent sequence and it have not convergent subsequence

Answer 1

I don't think that's possible, just by looking at the definition of a convergent sequence:\[\lim_{n \rightarrow \infty} a_n=L~~\text{means that}~(\forall \epsilon >0)~(\exists N\in\mathbb{N}):\forall n\geq N,~|a_n-L|<\epsilon.\]So any subsequence has a lower and an upper bound which would both tend to L. So there is no example.

@eliassaab

Answer 2

i got the answer..... thank

Answer 3

Any subsequence of a convergent sequence is itself convergent.