What is this convergence theorem?

Question

Let $w_k$ be a Cauchy sequence of functions continuously differentiable in the closure of $D$. By Schwarz inequality, $$\left[ \int_D (w_k - w) \phi dx\right]^2 \le \int_D (w_k - w)^2 dx \int_D \phi^2 dx.$$ Then we have $$\lim_{k\to \infty} \int_D w_k \phi dx = \int_D w \phi dx,$$ for any trial function $\phi$ with an integrable square.