# real analysis – closed convex hull with weak convergent sequence

The $$H$$ to be a space of Hilbert and Suppose $$x_n$$ converges weakly towards $$x$$ in $$H$$. Let $$K_n$$ to be the closed convex hull $$bar {co} {x_k: k geq n }$$. I would like to show $$bigcap K_n = {x }$$.

What I know so far is that for a convex set, the weak closure is the same as closing the norm in $$H$$. $$x$$ is clearly in $$K_n$$because it is in the weak closing of the tail $${x_k: k geq n }$$ for each $$n$$. But how can I show that $$x$$ is the only element in $$bigcap K_n$$?