I've got an exercise to prove Lax-Milgram theorem using Galerkin Approximation Method. Below I present statement of theorem from my lectures and my try.
Theorem Let $H$ be real Hilbert space and let $B: H \times H \to \mathbb{R}$ be bilinear form which is bounded i.e.: $$ \exists \alpha > 0 \ \ \forall u,v \in H \ \ |B(u,v)| \leq \alpha ||u||_H ||v||_H$$ and coercive i.e.: $$ \exists \beta > 0 \ \ \forall u \in H \ \ B(u,u) \geq \beta ||u||_H^2.$$ If $f: H \to \mathbb{R}$ is linear and bounded functional then there exists unique $u \in H$ that $$ B(u,v) = \langle f, v \rangle,$$ where $\langle \cdot, \cdot \rangle$ means value $f$ at $u$.
My try Let $\{w_k\}$ be the orthonormal base in $H$ and let $u_m = \sum_{k=1}^{m} d_m^k w_k$ (it is $m$-th Galerkin Approximation of $u$). Now we've got: $$B(u_m, w_k) = \langle f, w_k \rangle, \ \ \ k = 1, ..., m$$ and then $$ \sum_{j=1}^m d^j_m B(w_j, w_k) = \langle f, w_k \rangle$$. Now I would like to prove that this system of equations has unique solution by Cramer theorem (because det$[B(w_j, w_k)]_{j,k=1,...,m} \not= 0]$ from ellipticity. So we've got $$ B(u_m, u_m) = \langle f, u_m \rangle $$ and I have no idea how to go to the limit. I will be very greatful for help.