Condition for finite $\infty$-norm of a transfer function

Question

I am reading "Feedback control theory" by Doyle, Francis and Tannenbaum DFT.

Lemma 1 on page 16 states: the $\infty$-norm of a transfer function $G$ is finite iff $G$ is proper and has no poles on the imaginary axis.

I am wondering why stability of $G$ doesn't come into the picture here.