  Let $G$ be a real linear reductive group and $K$ be a maximal compact
subgroup. Let $P$ be a minimal parabolic subgroup of $G$ with complexified Lie
algebra $\mathfrak{p}$, and $\mathfrak{n}$ be its nilradical. In this paper we
show that: for any admissible finitely generated moderate growth smooth
Fr\'echet representation $V$ of $G$, the inclusion $V_{K}\subset V$ induces
isomorphisms $H_{i}(\mathfrak{n},V_{K})\cong H_{i}(\mathfrak{n},V)$ ($i\geq
0$), where $V_{K}$ denotes the $(\mathfrak{g},K)$ module of $K$ finite vectors
in $V$. This is called Casselman's comparison theorem. As a consequence, we
show that: for any $k\geq 1$, $\mathfrak{n}^{k}V$ is a closed subspace of $V$
and the inclusion $V_{K}\subset V$ induces an isomorphism
$V_{K}/\mathfrak{n}^{k}V_{K}= V/\mathfrak{n}^{k}V$. This strengthens
Casselman's automatic continuity theorem.
