Let's see... Let be the Galerkin approximation. Identify elements in each subspace with their natural injections into . Coercivity of implies that , which gives that is bounded in and must therefore possess a subsequence, denoted by also, to weakly converge to . Since is linear, we have which means is also a solution of the initial problem. Now, to demonstrate the strong convergence of , notice that coercivity and Galerkin orthogonality imply .