Solution
= Solution
For separable Banach $X$, the closed unit ball of $X^*$ 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>.