Let $\Omega$ be a polygonal domain in $\mathrm{R}^2$ and let $\mathcal{T}_0, \mathcal{T}_1,...$ be a sequence of triangulations of $\Omega$ formed by standard refinement. Let $\mathcal{P}_k^1$ be the space of continuous piecewise linear functions relative to $\mathcal{T}_k$. Assume that $u\in H^1(\Omega)$ (but $u$ is not necessarily in $H^2(\Omega))$ and let $u^{(k)}$ be the best approximation to $u$ from $P_k^1$ in the $H^1(\Omega)$-norm. Please can you help me to prove that
$$\|u-u^{(k)}\|_{H^1(\Omega)}\longrightarrow 0$$
What I tried to do is that since the space $H^2(\Omega)$ is dense in $H^1(\Omega)$, there exist a sequence $\{v_j\}$ such that $$\|u-v_j\|_{H^1(\Omega)}=0$$ Let $v_j^{(k)}$ be the best approximation to $v_j$ from $\mathcal{P}_k^{(1)}$, then for each $j$, $$\|u-u^{(k)}\|_{H^1(\Omega)}\leq\|u-v_j^{(k)}\|_{H^1(\Omega)}$$
I do not know how to proceed from here.
