 Solution

For separable Banach $X$, the closed unit ball of $X^{\ast }$ is weak-star compact. Embed it in the product of compact discs by evaluations on a countable dense subset; diagonal subsequences converge on that subset, boundedness extends the limit to all $X$, and the limit functional remains in the ball. Metrizability of the bounded weak-star topology turns sequential compactness into compactness. Scaling gives the general Banach-Alaoglu theorem.