Can we say that $ \{x_n \} $ admits a convergent subsequence?

Question

Let $X$ be a banach space with separable dual $X^{*}$.

Let $\{x_n\}$ be a bounded sequence of $X$.

Can we say that $ \{x_n \} $ admits a convergent subsequence ?

Answer 1

This is false in general. Consider $X = \ell_2$. Then $X^\ast$ is separable and the sequence $(e_n)$ with

$$e_n = (\dots, 0, 0, 1, 0, 0, \dots)$$

(with $1$ on the $n$-th position) is a bounded one which clearly does not have a convergent subsequence.