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 $?