From Convex Optimization by Boyd & Vandengergh:
Let $||\cdot\|$ be any norm. Then $\|x\|_* \ge \gamma \|x\|_2$ for some $\gamma \in (0,1] $.
I start by assuming that $\gamma \gt 1$. Then
$$\gamma \|e_i\|_2 = \gamma \cdot 1 = \gamma \le \sup \{e^Tz : \|z\| \le 1\} = \sup \{z_i : \|z\| \le 1\}$$
where $e_i = \langle0, \dots, 0, 1, 0 , \dots, 0\rangle$, i.e., a single $1$ at position $i$ and $0$ everywhere else. From here I am having trouble showing a contradiction. Is there a simple next step or is there a completely easier way to show this?