Exercise 3.1.23 in Salamon's Functional Analysis

Question

I'm reading the book Functional Analysis written by Dietmar A. Salamon.The following problem is the exercise 3.1.23 in the book.

Let X be a Banach space and suppose the dual space of X is separable.Let S be a bounded subset of X and let x be an element in the weak closure of S.

I want to prove that there is a sequence in S that converges weakly to x. I don't know how to use the condition that X is a Banach space.

Any hint would be helpful,thanks a lot!