Proof that $H^{1}(\mathbb{R}^{2}) \subset L^{p}(\mathbb{R}^{2})$ for $2 \le p < \infty$

Question

I am trying to prove that $H^{1}(\mathbb{R}^{2}) \subset L^{p}(\mathbb{R}^{2})$ in dimension $d=2$ for $2 \le p < \infty$. In other words, I need to prove that for every $f \in H^{1}(\mathbb{R}^{2})$ and $2 \le p < \infty$ there exists a constant $C > 0$ depending only on $p$ such that: $$\|f\|_{L^{p}} \le C\|f\|_{H^{1}} = C(\|f\|_{L^{2}} + \|\nabla f\|_{L^{2}}).$$ I searched all over the internet to find materials on this inequality but Sobolev spaces are such a rich subject and I found all kinds of inequalities but this one. Any help with the proof or suggestion of where I can find its complete proof is very welcome.