Let $H$ be a Lie subgroup of a Lie group $G$. In Tu's book on manifolds he states
...being an immersion, the inclusion map $i: H \hookrightarrow G$ of a Lie subgroup $H$ is of course $C^\infty$.
This seems obvious, but I cannot see why the implication holds. Since $H$ is an immersed submanifold it may not have the same topology as $G$, so why should we expect continuity let alone smoothness?