I want to prove that every dual space is weak*-sequentially complete.
Let $X$ be a normed linear space and let $(f_n)$ be a weak* Cauchy sequence in $X^*$. Thus for all $x\in X$, $(f_n(x))$ is a Cauchy sequence in $\mathbb K$. Thus for all $x\in X$, $\lim\limits_{n\to \infty}f_n(x)$ exists. If I define $f(x)=\lim\limits_{n\to \infty}f_n(x)$ for all $x\in X$, then $f$ is linear. But how to show that it is bounded. Had $X$ been given a Banach space I could have done it by using Banach-Steinhauss theorem. But now how to proceed? Any hint is appreciated.
Hint : You can use the following corollary of Banach-Alaoglu Theorem :
Let $ X $ be a normed space. Then every bounded subset of $ X^* $ is relatively weakly$^*$ compact.