Let $M$ be a complete Riemannian manifold, $N\subset M$ a closed submanifold, $p\in M\setminus N$ and $d(p,N)$ the distance from $N$ to $p$. I would like to show that there exists a point $q\in N$ such that $d(p,q)=d(p,N)$.
Question: does there always exist a Cauchy sequence $(q_i)_{i\in \mathbb{N}}\subset N$ such that $d(p,q_i)\to d(p,N)$?
If so, the completeness of $M$ (and thus of $N$) would imply that the sequence $(q_i)_{i\in \mathbb{N}}$ converges to some $q\in N$ such that $d(p,q)=d(p,N)$, as desired.
How do I find such a sequence? Is the existence a property of complete metric spaces?
Let $q\in N$ and $G$ be a closed geodesic ball with center at $p$ and radius $r=d(p,q)$. The intersection $A$ of $G$ and $N$ is compact and consequently the function $d(p,q_i)$ restricted to $A$ attains its minimum in $A$.