A question on weakly convergence and norm convergence.

Question

Let $2 \le p<\frac{2n}{n-2}$. Suppose that a sequence $\{u_k\}_k\subset H^1(\mathbb{R}^n)$ weakly converges to $u \in H^1(\mathbb{R}^n)$, and hence weakly converges to $u$ in $L^p(\mathbb{R}^n)$. How can I prove that $$ ||u_k||_{H^i}-||u_k-u||_{H^i} \rightarrow ||u||_{H^i}, \ \ \ \ \ ||u_k||_{L^p}-||u_k-u||_{L^p} \rightarrow ||u||_{L^p} $$ as $k$ increases?