Proof that $\|fx\| \leq \|f\|\cdot\|x\|$
The inequality is clearly true if $x=0$. If $x\not=0$ write $$x=||x||\dfrac{x}{||x||}=||x||\cdot y$$ where $y=x/||x||$ has norm 1. The definition of the operator norm gives that $$||fy||\leq ||f||$$ hence $$\left|\left|f\left(\dfrac{x}{||x||}\right)\right|\right|\leq ||f|| \Rightarrow ||fx||\leq ||f||\cdot||x||$$ by linearity.