Let $H$ a Hilbert space and let $\{H_n\}_n$ be a sequence of finite-dimensional subspaces of $H$, such that $H_{n}\supset H_{n-1} \quad \forall n \in \mathbb{N}$ and $\cup_n H_n$ is dense on $H$. Let $B:H\times H \to \mathbb{R}$ be a bounded bilinear form and suppose for any $f \in H'$ there is a unique sequence $\{u_n(f)\}_n \subset H$ such that: \begin{align} \label{problem1} u_n(f) \in H_n \quad, \quad B(u_n(f),v_n)=f(v_n) \quad \forall v_n \in H_n, \tag{*} \end{align} and $\|u_n(f)\|\leq C_0\|f\|\quad \forall n \in \mathbb{N}$, where $C_0>0$ is a constant independent of $n$ and $f$. Suppose that for any $f \in H'$ there is a unique $u(f)\in H$ such that: \begin{align} \label{problem2} B(u(f),v)= f(v) \quad \forall v \in H. \tag{**} \end{align} Show that \begin{align*} \lim_{n \to \infty} \|u(f)-u_n(f)\|\to 0 \quad \forall f \in H'. \end{align*}
My attempts: The uniqueness of the sequence $\{u_n(f)\}_n$ implies that $u_n(f)=P_nu(f)$, where $P_n:H \to H$ is the Galerkin projection (i.e the only solution for $B(P_nv,w_n)=B(v,w_n) \quad \forall w_n \in H_n$).
Also the well posedness of (**) implies that the linear operator $\mathbb{B}:H\to H$ induced by the bilinear form $B$ (i.e $B(u,v)=\langle \mathbb{B}u,v \rangle$) is a linear isomorphism. In particular, it satisfies the inf-sup condition \begin{align} \sup_{v \in H}\frac{B(u,v)}{\|v\|_H} \geq \alpha \|u\|_H \quad \forall u \in H, \end{align} where $\alpha>0$ and also \begin{align} \sup_{u \in H}~ B(u,v) > 0 \quad \forall v \in H,~ v \neq 0. \end{align} The well posedness of (*) implies the discrete inf-sup condition: \begin{align} \sup_{v_n \in H_n}\frac{B(u_n,v_n)}{\|v_n\|_H} \geq \alpha_n \|u\|_H \quad \forall u_n \in H_n, \end{align} where $\alpha_n>0$.
I would like to use the Cea's Lemma \begin{align} \|u_n(f)-u(f)\| \leq \{1+ \frac{\|\mathbb{B}\|}{\alpha_n}\}\inf_{z_n \in H_n} \|z_n-u(f)\|, \end{align} but the problem is that there needs to be a constant $\tilde{\alpha}>0$ which satisfies $\alpha_n\geq \tilde{\alpha} \quad \forall n \in \mathbb{N}$. So maybe we need another approach.
Any suggestion is welcomed.