I am currently trying to understand closest point projections to a given embedded submanifold $M$ of $\mathbb{R}^n$. In Lee's "Smooth manifolds" I read about the existence of tubular neighborhoods. So far, so good. However, I now stumbled over the following statement multiple times: Each point $u$ of a tubular neigborhood $U$ posseses a unique closest point $m$ in $M$.
I understand that if I there is a closest point $m$ to $u$, then $u-m$ has to be in $N_mM$, therefore the closest point has to be unique. However, what I do not see is why a closest point must exist without further assumptions on the submanifold like being closed or compact. Why is this the case? Probably, there is a very easy argument I am missing, as I cannot see anyone talk about my issue...