I have to prove that if T : V $\displaystyle \rightarrow$ V and S:V$\displaystyle \rightarrow$V are diagonalizable so T$\displaystyle \otimes$S is diagonalizable.
Choose an eigenbasis for each $\displaystyle T,S$ and then form the standard tensor basis--I'm sure you'll find this is an eigenbasis for $\displaystyle T\otimes S$.