How can I prove that $\|Ah\| \le \|A\| \|h\|$ for a linear operator $A$? On http://www.proofwiki.org/wiki/Definition:Norm_(Linear_Transformation) , it is stated that $||Ah|| \leq ||A||||h||$ where $A$ is an operator.
Is this a theorem of some sort? If so, how can it be proved? I've been trying to gather more information about this to no avail.
Help greatly appreciated!

Note, that if $\|h\| = 0$ then the inequality trivially holds. Otherwise,
$$
  \frac{\|Ah\|}{\|h\|}\leq \sup_{\|h\|>0}\frac{\|Ah\|}{\|h\|}
$$
and thus
$$
  \|Ah\|\leq \|h\|\cdot\sup_{\|h\|>0}\frac{\|Ah\|}{\|h\|}
$$
By one of the (equivalent) definitions you linked,
$$
  \|A\| = \sup_{\|h\|>0}\frac{\|Ah\|}{\|h\|}
$$
