Show for any induced matrix norm and nonsingular matrix A that
$$
\left\|A^{-1}\right\| ≥ (\left\|A\right\|)^{-1}
$$
where
$$
\left\|A^{-1}\right\| = \max_{\left\|x\right\|=1}\{\left\|A^{-1}x\right\|\}\\
\left\|A\right\| = \max_{\left\|x\right\|=1}\{\left\|Ax\right\|\}.
$$
I am not sure how to show that:
\begin{equation}
\begin{split}
\left\|A^{-1}\right\| ≥ (\left\|A\right\|)^{-1}\\
\text{or}\\
\max_{\left\|x\right\|=1}\{\left\|A^{-1}x\right\|\} ≥ (\max_{\left\|x\right\|=1}\{\left\|Ax\right\|\})^{-1}
\end{split}
\end{equation}
Best Answer
Use $\lVert AB\rVert \leq \lVert A\rVert \lVert B\rVert$, as the induced norm is in particular submultiplicative. So that $\lVert I_n\rVert \leq \lVert A\rVert \lVert A^{-1}\rVert$.