Does a collar along the boundary always find an embedding in the manifold?

Question

Suppose we have a smooth manifold $M$ with boundary $K$. $K$ always finds a natural embedding inside $M$. Can we extend this embedding to an embedding of $K\times [0,\epsilon)$ inside $M$ in general ? It seems to be the case in all examples that I can think of, but I can not find a proof.

Answer 1

A search for "collar neighborhood theorem" led to this, in which Theorem 1.0.5 is the theorem you want (assuming you're talking about smooth manifolds with smooth boundaries, which I'm assuming because you've put "differential geometry" as a tag).

I think that there's also a nice proof of this in Milnor's Lectures on the h-cobordism Theorem.