Existence of a solution to the Dirichlet problem

Question

Let $\Omega\subseteq \mathbb{R}^N$ be a regular domain (a bounded open set with $C^1$ boundary). Let $g\in H^{\frac12}(\Omega)$ and let $$\DeclareMathOperator{\Dm}{d\!} H^1_g(\Omega):=\{u\in H^1(\Omega):\text{Tr}(u)=g\}$$ where $\text{Tr}$ is the trace operator. The Dirichlet Problem consists in finding which $u\in H^1_g(\Omega)$ minimize the following integral $$ F(u):=\int_{\Omega} \left[\frac{|\nabla u|^2}{2}-fu\right]\Dm x$$ where $f$ is a fixed $L^2(\Omega)$ function. I already know that if a solution exists then it's unique, but I'm having some difficulties in the existence proof.

My attempt of understanding the proof

Let $\{u_n\}\subseteq H^1_g(\Omega)$ be a minimizing sequence, i.e. a sequence such that

$$F(u_n)\to \inf_{u\in H^1_g(\Omega)} F$$

We just have to prove that $\{u_n\}$ is bounded in $H^1(\Omega)$ (because then it admits a weakly convergent subsequence and I understand well that its limit is a solution). Let $G$ be a $H^1(\Omega)$ function such that $g=\text{Tr}(G)$. I can define $v_n:=u_n-G\in H^1_0(\Omega)$ and now I just need to prove $\{v_n\}$ is bounded in $H^1(\Omega)$.

The proof proceeds using the following inequalities $$ \begin{split} \|v_n\|_{H^1(\Omega)}^2 &\leq C \|v_n\|^2_{L^2(\Omega)} \\ & \leq C (\|\nabla u_n\|_{L^2(\Omega)}^2+\|\nabla G\|_{L^2(\Omega)}^2) \\ & \leq C_1(1+\|u_n\|_{L^2(\Omega)}) \leq C_2(1+\|v_n\|_{L^2(\Omega)})\leq C_2(1+\|v_n\|_{H^1(\Omega)}) \end{split} $$ I understand just few of these 5 inequalities. I'll comment them in order

1. Clearly $$ \|v_n\|_{H^1(\Omega)}^2=(\|v_n\|_{L^2(\Omega)}+\|\nabla v_n\|_{L^2(\Omega)})^2\leq (K\|\nabla v_n\|_{L^2(\Omega)}+\|\nabla v_n\|_{L^2(\Omega)})^2$$ where I simply used Poincarè inequality in the last step.

2. Triangular inequality.

3. I don't know. I think my notes suggest that we should somehow use the fact that $$ \frac{1}{2}\|\nabla u_n\|^2_{L^2(\Omega)}-\|f\|_{L^2(\Omega)}\|u_n\|_{L^2(\Omega)}\leq F(u_n)\leq F(G)+1. $$ I perfectly understand these inequalities but I don't know how to use them.

4. I think that we are simply using $\|u_n\|_{L^2(\Omega)}\leq \|v_n\|_{L^2(\Omega)}+\|G\|_{L^2(\Omega)}$ so if $C_2=C_1+\|G\|_{L^2(\Omega)}$ everything holds but I'm not really sure. Am I right?

5. Obviously $\|v_n\|_{L^2(\Omega)}\leq \|v_n\|_{H^2(\Omega)}$.

Thank you in advance. I'm trying to reconstruct the proof of the course so I'm NOT primarily interested in alternative approaches.