Tubular neighborhood theorem

This fact is an application of the following pivotal fact/result/idea: inverse function theorem
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This fact is an application of the following pivotal fact/result/idea: existence of smooth partitions of unity
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This fact is an application of the following pivotal fact/result/idea: Lebesgue number lemma
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Statement

Let $M$ be a submanifold (differential sense) of $\R^n$, of dimension $m$. Then, there exists $\epsilon>0$ such that for any point at distance at most $\epsilon$ from $M$, there is a unique expression of the point as a sum $p + v$ where $p \in M$ and $v$ is a normal at $p$, with $\| v \| < \epsilon$.

If we define $U$ as the open subset of $\R^n$ comprising those points of $\R^n$ at distance less than $\epsilon$ from $M$, then $U$ can be viewed as a concrete realization, in the ambient space $\R^n$, of the normal bundle to $M$ in $\R^n$. In the situations where the normal bundle to $M$ is trivial, we see that this gives a natural diffeomorphism $U \cong M \times \R^{n-m}$.