How to prove these equivalences?

Question

I want to prove the following statement:

Let $K$ be a compact Hausdorff space and $F\subset C(K)$. Then the following are equivalent:

1. The closure of $F$ in the weak topology of $C(K)$ is weakly compact.
2. $F$ is bounded and its closure in the topology of pointwise convergence is compact.
3. $F$ is weakly sequentially compact.

I already proved that $(1\Rightarrow 2)$. The implication $(3\Rightarrow 1 )$ is due to Eberlein Theorem. The problem is Proving that $(2 \Rightarrow 3)$.

Another way to prove the equivalence would be to prove that $( 1\Leftrightarrow 2)$. From Eberlein-Smulian Theorem we have already have that $( 1\Leftrightarrow 3)$. But even when trying to prove $(2\Rightarrow 1)$ I face problems. Can somebody give me any hint how to prove the equivalence? Thank you in advance! :)