Poisson's equation

Question

By definition $\{ {w_k}\} _{k = 1}^\infty \ $ is an orthorgonal basis in $H_0^1(U)$ and an orthornomal basis in $L^2(U)$.

suppose $f \in {L^2}(U)$ and assume that ${u_m} = \sum\limits_{k = 1}^m {d_m^k{w_k}}$ solves

$$\int\limits_U {D{u_m} \cdot D{w_k}dx = } \int\limits_U {f \cdot {w_k}dx} $$ for $k=1,2,\cdots\ m$.

Show that a subsequence of $\{ {u_m}\} _{m = 1}^\infty $converges weakly in $H_0^1$ to the weak solution of

$$\left\{\begin{align*} &- \Delta u = f&&\text{in }U\\ &u = 0&&\text{on }\partial U \end{align*} \right.$$

Actually I know that we can according to the usual energy estimation to obtain the subsequence, and integrate by parts,while I still don't konw how to pass to limit to in the integrating equation.Wish to somebody to give the details, Thanks a lot!