Submanifold is complete

Question

If $M$ is a complete manifold and $N\subset M$ is a closed, embedded submanifold with the induced Riemannian metric, show that $N$ is complete.

I really don't know where to start. This is not homework, please help! Thank you very much.

Answer 1

First, the intrinsic topology of $N$ agrees with the subspace topology of $N$. Next, you need to observe that the Riemannian distance functions $d_N, d_M$ are related (on $N$) by the inequality $$ d_N\ge d_M. $$ Now, apply the Cauchy criterion for completeness as well as the fact that $N$ is closed in $M$.