Let $(N,h)$ be a compact Riemannian submanifold of the Euciledean space $\mathbb{E}^q$. i.e.$N$ is a compact submanifold of $\mathbb{E}^q$ and $h$ is the Riemannian metric of $N$ derived from the metric of $\mathbb{E}^q$. Then, is the following statements true? How to prove it?
\begin{eqnarray} \exists \varepsilon>0\ \ \ s.t. \ \ \ d(x,N)<\varepsilon\Rightarrow \exists! y\in N\ \ \ |y-x|=d(x,N) \end{eqnarray}
Yes, this is the content of the $\epsilon$-Neighborhood Theorem, and it also guarantees that the map from the $\epsilon$-neighborhood of $N$ to $N$ assigning each point the nearest point in $N$ is a submersion. Another version of the theorem allows $N$ to be noncompact at the cost of allowing $\epsilon$ to depend on the point in $N$.
Even in the compact case the proof is slightly technical, but it amounts to identifying the fibers of the normal bundle of $N$ with planes in $\mathbb{E}^q$ in the obvious way and then restricting attention to balls in each plane small enough that they don't overlap. The compactness of $N$ allows us to choose "small enough" uniformly. For a detailed and reasonably clear proof see Guillemin & Pollack's Differential Topology:
http://books.google.com.au/books?id=FdRhAQAAQBAJ&lpg=PA218&ots=hdu6edyn-d&dq=epsilon%20neighborhood%20theorem&pg=PA69#v=onepage