Let $X$ be a separable infinite-dimensional Banach space.
Assume that every element of $B_{X^{**}}$ is the weak-star limit of a sequence in $B_{X}$.
Clearly, by the canonical embedding, we can see that $\text{card} (X) \leq \text{card}(X^{**})$.
If we choose a norm-dense sequence $\{x_n\}_{n=1}^{\infty}$ in $B_{X}$, then from our assumption, I could check that each element of $B_{X^{**}}$ is the weak-star limit of a subsequence of $\{x_n\}_{n=1}^{\infty}$.
Here is my question: How can I conclude that $\text{card} (X) = \text{card}(X^{**})$ from the above observation?
Any help will be appreciated.