Show that a weakly compact set in a normed space is norm-closed and norm-bounded. Suppose $X$ is a normed space and $K$ is a subset of $X$ such that $K$ is weakly compact. Show that $K$ is norm-closed and norm-bounded.
I manage to show that $K$ is norm-bounded by using the Uniform Boundedness Theorem. However, I am not sure on how to show that $K$ is norm-closed. The following is my attempt:
Since $K$ is weakly compact, we have $x^*(K)$ is compact in the scalar field of $X$, say $F$. By the Heine-Borel Theorem, $x^*(K)$ is closed. Since $x^*$ is continuous, we have $K$ is norm-closed. 
Is it correct?

No, your reasoning is not correct. In fact, you only have shown that the preimage of $x^*(K)$ w.r.t. $x^*$ is closed. But
$$K \subset (x^*)^{-1}(x^*(K))$$
might be a proper inclusion. Even
$$K \subset \bigcap_{x^* \in X^*} (x^*)^{-1}(x^*(K))$$
might be proper.
However, the argument for closedness is quite simple. Assume that $K$ is not closed. Hence, there is a sequence $\{x_n\}$ such that ... Using the weak compactness of $K$ we find...
