Weakly sequentially compact subspace is closed?

Question

Let $E$ be a Banach space equipped with the weak topology $\sigma(E,E^\star)$. I know that a weakly sequentially compact subspace is weakly closed: in fact by Eberlein-Smulian theorem it is weakly compact and therefore also weakly closed (since the topology $\sigma(E,E^\star)$ is Hausdorff). Is there a simple proof of this implication that does not use Eberlein Smulian theorem (since I want to prove this fact to prove a simpler version of Eberlein Smulian theorem)? Thank you!