Show that $\frac{||r||_2}{||A||_2}\leq||z-x||_2\leq||A^{-1}||_2||r||_2$

Question

Let $A\in\mathbb{R}^{n\times n}$ be nonsingular and let $Ax=b$ and $r=Az-b$ for some z. Show that $\frac{||r||_2}{||A||_2}\leq||z-x||_2\leq||A^{-1}||_2||r||_2$.

Working backwards I got: $||A^{-1}||_2||r||_2\geq||A^{-1}r||=||z-A^{-1}b||_2=||z-x||_2$. Any ideas on how to obtain the last inequality?

Answer 1

$$\|r\| = \|Az-b\| = \|A(z-x)\| \le \cdots$$