weak convergence in an unit ball.

Question

let $x_n \rightarrow x$ weakly and $x_n \in \overline{B(0,1)}$ for all $n$ and $\|x\|_H = 1$ then $\|x_n-x\|_H \rightarrow 0$. We can write $\|x_n -x \|_H^2 = <x,x>+<x_n,x_n> -2Re<x_n,x> = 0$ when i apply the weak convergence assumption. Not sure if my argument is correct since, i did not use that assumption that $x_n \in \overline{B(0,1)}$ for all $n$ and $\|x\|_H = 1$.