Given an induced matrix norm $||\cdot||$ such that $\exists_{\varepsilon >0} \forall_{x\in\mathbb{R} ^n} ||Ax||\ge\varepsilon ||x||$ prove that $||A^{-1}||\leq\frac1\varepsilon$.
I figured out that $||A||\ge \varepsilon$ and that $\varepsilon \varepsilon^{-1}=1 =||AA^{-1}||\le||A||||A^{-1}||$, but however I manipulate those terms, I never can get the inequality right.
Abridged solution. Set $y = A(x)$ so that $x = A^{-1}(y)$ and so the hypothesis $\|A(x)\|\geq \varepsilon \|x\|$ is the desired conclusion. QED