# Convergence of a sequence by the convergence of a subsequence

Let $$X$$ be a Banach space and $$\{x_{n}\}_{n\in\mathbb{N}}$$ be a bounded sequence in $$X$$. Assume for any subsequence $$\{x_{n_{k}}\}_{k\in\mathbb{N}}$$ of $$\{x_{n}\}_{n\in\mathbb{N}}$$, there exists a subsequence $$\{x_{n_{k_{l}}}\}_{l\in\mathbb{N}}$$ of $$\{x_{n_{k}}\}_{k\in\mathbb{N}}$$ which converges to $$x$$ in $$X$$. Can I claim that $$x_{n}\to x$$ in $$X$$?

I know that if every subsequence of a sequence converges then the sequence converges but I dont know whether this holds true or not for the subsequence of a subsequence. My intuition tells me that it is not true but I cannot get a good counterexample.

It is true and yo u can prove it by contradiction. If it is not true that $$x_n \to x$$ then there exist $$\epsilon >0$$ and integers $$n_1 such that $$\|x_{n_i}-x\| \geq \epsilon$$ for all $$i$$. Can you see from this that $$\{x_{n_i}\}$$ cannot have a subsequence converging to $$x$$?.