Induced 1 Norm Proof

Induced 1 Norm Proof

I was trying to prove the expression for the Induced 1 norm but was unable
to. Is there a simpler proof than norm of $T$ induced by the 1-norm?
I began by trying the following: \begin{eqnarray*} \max & & \|Ax\|_{1}\\
\|x\| & \leq & 1\\ & \downarrow\\ \max & & \sum_{m}|\sum_{n}A_{i}x_{i}|\\
\|x\| & \leq & 1\\ & ? \end{eqnarray*}