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 $n_1<n_2<...$ 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 $x$?.