During the proof of the fact that every smooth submanifold $N \subset R^n$ has a tubolar neighborhood, I need to prove that there is a smooth function $f: N \mapsto \mathbb{R}^+$ that "squeezes" the tubolar nbhd into an open set $U$ such that the map $g:(p,v) \mapsto p+v$ is an embedding from $U$ to $\mathbb{R}^n$. Now, I get why this works locally ($dg_{(p,0)}$ is the identity, if I am not mistaken), but I would like to see a relatively rigorous proof that this "squeezing function" is smooth and more important that $g$ restricted to $U$ is an embedding. Again, this sounds obvious given how $g$ is defined, but I would like to see a sufficiently rigorous proof.
EDIT: As pointed out, it should be explained why I bother searching for such $f$. The fact is that $g$ is an embedding if you restrict yourself on a sufficiently small neighborhood $U$ of $N$ seen into its normal bundle. The squeezing function is required to embed $\nu N$ into $U$ leaving $N$ fixed. This way I’d have that $g \circ f$ is an embedding of $\nu N$ in $\mathbb{R}^m$ that is also an open nbhd of $N$ and leaves $N$ fixed, so it is a tubolar nbhd.