closed convex hull of weak convergent sequence

Since $x_n$ converges weakly to $x$, one has for every $a \in H$. $$\lim_{n \to \infty} \langle a, x_n \rangle = \langle a, x \rangle.$$ Consequently, one has $$\lim_{n \to \infty} \sup_{z \in K_n} |\langle z, a\rangle - \langle x, a \rangle| = 0.$$ This implies for $y \in \bigcap K_n$, one has for every $a \in H$ $$\langle y, a \rangle = \langle x, a \rangle,$$ which yields $y = x$.