Lemma 1

Let G be a locally compact group, $\rho$ a lower semi-continuous finite numerical function $\geqslant 0$ on G such that $\rho(st) \leqslant \rho(s)\rho(t)$ for all $s,t \in G$. Then $\rho$ is bounded above on every compact subset of G.

There exists a nonempty open subset U of G such that $\rho$ is bounded above on U (GT, IX, §5, No. 4, Th. 2). Let K be a compact subset of G. Then K is covered by a finite number of sets $s_1U, \ldots, s_nU$. For every $x \in U$, one has $\rho(s_ix) \leqslant \rho(s_i)\rho(x)$, therefore $\rho$ is bounded above on the $s_iU$, hence on K.