lp spaces - Proof of infinity matrix norm

Given the $l_{\infty}$ matrix norm for $A{\in}{\Bbb{R}}^{mxn}$ is defined as: $\|A\|_{\infty} =\max_{1 \leq i \leq n}\|a^{i}\|_{1}$ (where $a^{i}$ is the i$^{th}$) row in matrix A),

Show that:
$\|A\|_{\infty} =\max \left\{\|Ax\|_{\infty} : x_{\infty} \le 1\right\} =\max \left\{\|Ax\|_{\infty} : x_{\infty} = 1\right\}$

I know that this is a property of subordinate matrix norms but I'm not sure how to go about with proving it.