Proving that $\|A\|$ is finite. Let $|v|$ be the Euclidean norm on $\mathbb{R^n} $. For  $A\in \mathrm{Mat}_{n\times n}(\mathbb{R})$ we define $\displaystyle \|A\|:= \sup_{\large v\in \mathbb{R^n},\,v \neq 0}\frac{|Av|}{|v|}$. How to show that $\|A\|$ is finite for every $A$?
It would be very helpful if someone could give hints. I think I should show that $\|-\|$ is bounded,but I don't know how...

Hint: 
Let $S=\{v\in\mathbb{R}^n\;|\;|v| = 1\}, N = \{\frac{|Av|}{|v|}\;|\;v\in\mathbb{R}^n.\;v\ne 0\}, N' = \{|Av|\;|\;v\in\mathbb{R}^n.\;|v|=1\}$  
Step 1: $||A|| = \sup N = \sup N'$
Step 2: Show that $x\to|Ax|$ is a continuous map. $S$ is closed and bounded in $\mathbb{R}^n$ therefore compact,  |Ax| attains max on $S$. 
Done
Do you happen to be in Linear Algebra II class?
