Matrix Norm Lemma

Question

There is a lemma claims that : $||Ax||/||x|| \le max_{||x||\ne 0} (||Ax||/||x|) = ||A|| $

I'd like to know how come $||Ax||/||x|| \le max_{||x||\ne 0} (||Ax||/||x|)$ because it does not make sense to me.

Thanks in advance.

Answer 1

Note that

$$\|A\| := \sup\{ \|Aw\| : \|w\| = 1\} \geq \|Ay\|$$

for all $y$ with $\|y\| = 1$. But for any $x \neq 0$, we know that $y:= \dfrac{x}{\|x\|}$ has norm $1$. So substituing this choice of $y$ into the above inequality gives the required result.