How can I show $||v||^{2}_{2} \leq ||v||_{1} \cdot ||v||_{\infty}$ holds? That is, for a vector in $\mathbb{R}^{n}$,
$$\sum_{i=1}^{n}|v_i|^{2} \leq \max_{i=1}^{n} |v_{i}| \cdot \sum_{i=1}^{n} |v_i|?$$
I have managed to prove the result that $||v||_{\infty} \leq ||v||_{2}$ holds; I don't know if that helps in any way. I was thinking of writing $||v||_1 \cdot ||v||_{\infty} \geq ||v||_{1} \cdot ||v||_{2}$, but I didn't get anywhere.
Slightly elaborating on @d.k.o.'s comment: \begin{align} \sum_{i=1}^n | v_i |^2 &= \sum_{i=1}^n |v_i| |v_i| \\ \tag{1}&\leq \sum_{i=1}^n \| v \|_\infty | v_i | \\ &= \| v \|_\infty \sum_{i=1}^n |v_i| \\ &= \|v\|_\infty \|v\|_1. \end{align} In step (1) we used the fact that $| v_i | \leq \| v\|_\infty$.