Show $\vert \vert u \vert \vert_{H^k(\Omega)} \leq C[\vert \vert u \vert \vert_{L^2(\Omega)} + |u|_{H^k(\Omega)}]$ using Deny and Lions lemma

Question

Let $u \in H^k(\Omega)$ where $\Omega$ is a bounded Lipschitz domain. I want to show \begin{equation} \vert \vert u \vert \vert_{H^k(\Omega)} \leq C[\vert \vert u \vert \vert_{L^2(\Omega)} + |u|_{H^k(\Omega)}] \end{equation} where $|u|_{H^k(\Omega)}^2 = \sum_{|\alpha|=k}\vert\vert D^{\alpha}u \vert\vert^2$ and $\vert \vert u \vert \vert_{H^k(\Omega)}^2 = \sum_{|\alpha| \leq k}\vert \vert D^{\alpha}u \vert \vert^2$ using Deny-Lions lemma \begin{equation} \inf_{v \in \mathcal{P}_{k-1}}\vert \vert u - v \vert \vert_{H^k(\Omega)} \leq C|u|_{H^k(\Omega)} \end{equation} where $\mathcal{P}_{k-1}$ is the space of all polynomials of degree $k-1$. So far I got \begin{equation} \vert \vert u \vert \vert_{H^k(\Omega)} \leq \vert \vert u - v \vert \vert_{H^k(\Omega)} + \vert \vert v \vert \vert_{H^k(\Omega)} \end{equation} for all $v \in \mathcal{P}_{k-1}$. Therefore \begin{equation} \vert \vert u \vert \vert_{H^k(\Omega)} \leq \inf_{v \in \mathcal{P}_{k-1}}\left( \vert \vert u - v \vert \vert_{H^k(\Omega)} + \vert \vert v \vert \vert_{H^k(\Omega)}\right) \end{equation} which doesn't help much. From this I don't know how to proceed. Also I get \begin{align} \vert \vert u \vert \vert_{H^k(\Omega)}^2 &= \vert \vert u \vert \vert_{L^2(\Omega)}^2 + \sum_{1 < |\alpha| < k}\vert \vert D^{\alpha}u \vert \vert_{H^k(\Omega)}^2 + |u|_{H^k(\Omega)}^2 \\ &\leq \vert \vert u \vert \vert_{L^2(\Omega)}^2 + |u|_{H^k(\Omega)}^2 + \sum_{1 < |\alpha| < k}\vert \vert D^{\alpha}(u-v) \vert \vert_{H^k(\Omega)}^2 + \sum_{1 < |\alpha| < k}\vert \vert D^{\alpha}v \vert \vert_{H^k(\Omega)}^2 \end{align} Now I can't see how to proceed.
Edit: It is possible to show by contraposition that there exists a constant $K > 0$ such that \begin{equation} ||u||_{H^k(\Omega)} \leq K [|u|_{H^k(\Omega)} + ||\Pi u||_{L^2(\Omega)}] \mbox{ for all } u \in H^k(\Omega) \end{equation} where $\Pi : L^2(\Omega) \to \mathcal{P}_{k-1}$ is the $L^2$-projection. From this we get \begin{align} ||u||_{H^k(\Omega)} &\leq K [|u|_{H^k(\Omega)} + ||\Pi u||_{L^2(\Omega)}] \mbox{ for all } u \in H^k(\Omega) \\ &\leq K \mbox{max}(1,||\Pi||)[|u|_{H^k(\Omega)} + || u||_{L^2(\Omega)}] \\ &= K'[|u|_{H^k(\Omega)} + || u||_{L^2(\Omega)}]. \end{align} If someone is interested I can write the proof in the answer section.