In order to prove a larger assumption, I need to find a vector norm over $M_n$ such that $\|I\| < 1$. None of the standard $p$-norms, nor the infinity norm work. I know that for matrix norms, this is impossible, but I think it should be possible with a vector norm since I don't have to worry about submultiplicativity.
Any hints would be greatly appreciated. Proving an example is in fact a norm is within my grasp, I just need some idea about how to construct an example.