Why the 2 norm of symmetric matrix A in bounded by its infinity norm?

Question

For an abstract matrix $A$ of dimension $p\times p$ and $p$ can approach infinity, we known that $\|A\|_2 \leq \sqrt{p}\|A\|_\infty$. However, in some papers, e.g., the sentences below A.18 in page 13, and the sentences above Lemma 3 in page 13, I see that when $A$ is symmetric, $\|A\|_2 \leq \|A\|_\infty$, how to prove it?