I need to show that if $\displaystyle \bf{M}$ is an $\displaystyle m \times n$ matrix, then $\displaystyle ||\bf{M}(\bf{h})||\leq \alpha ||\bf{h}||$ for some $\displaystyle \alpha$ by estimating the entries of $\displaystyle \bf{M}$. The hint I am given is that $\displaystyle \bf{M}(\bf{h}) = \bf{y} = (y_1, \ldots, y_m)$ where $\displaystyle y_i=\bf{m}_i \cdot \bf{h}$ for some $\displaystyle \bf{m}_i \in R^n$ and apply the CS inequality.

I have not done multivariable calc in a VERY long time, and I just can't seem to jumpstart my brain to attack this problem.