Let G be a real reductive group, and P any parabolic subgroup. In the paper 'Canonical extensions of Harish-Chandra modules to representations of $G$' by Casselman, a result says that if we begin with a smooth representation of moderate growth of P, then the smooth induced representation of G induced from P is also of moderate growth.(Proposition 4.1 in that paper)

Casselman didn't give details of the proof. Does anybody know any proof of this result in other places?