The operator norm on a net

Let $\Ncal$ be an $\eps$-net of $S^{d-1}$ with $\eps<1$. Prove that for any $A\in\R^{m\times d}$,

$$\sup_{x\in\Ncal}\|Ax\|_2 \le \|A\| \le \frac{1}{1-\eps}\sup_{x\in\Ncal}\|Ax\|_2 .$$

For a symmetric $A$, prove the analogous statement for the quadratic form with $\|A\| \le (1-2\eps)^{-1}\sup_{x\in\Ncal}|x^\top Ax|$ when $\eps<1/2$.