Show Matrix 2-Norm Greater than Matrix Inf-Norm

blabbate

1. The problem statement, all variables and given/known data
Show [tex]\sup _{x \neq 0}\|Ax\|_\infty/\|x\|_\infty\leq\|Ax\|_2/\|x\|_\infty[/tex]

2. Relevant equations
See above.


3. The attempt at a solution
I'm not looking for a full solution, just a push in the right direction. I've tried breaking out the numerators such that [tex]\|Ax\|\leq\|A\|\|x\|[/tex] on both sides and multiplying through, as well as trying to use [tex]\|x\|_2\geq\|x\|_\infty[/tex] and substituting to get a suitable inequality, but no luck.