Let $M$ be a topological manifold with boundary. I'm trying to prove that $M - \partial M \simeq M$, where "$\simeq$" denotes homotopy equivalence, using the collar neighbourhood theorem. This states that for a topological manifold with boundary $M$, there exists a diffeomorphism between $U$ and $\partial M \times [0,1)$, where $U \subset M$ is an open neighbourhood of $\partial M$. My approach is as follows:
We show that, $M = M \cup U \simeq M \cup (\partial M \times [0,1)) \simeq M \cup (\partial M \times (0,1)) \simeq M \cup (U - \partial M) = M - \partial M$.
By the collar neighbourhood theorem, we have a diffeomorphism, and so a homotopy equivalence between $U$ and $\partial M \times [0,1)$. From this we obtain the homotopy equivalence $M \cup U \simeq M \cup (\partial M \times [0,1))$.
I also already have a homotopy equivalence $[0,1) \simeq (0,1)$, from which we obtain the homotopy equivalence $\partial M \times [0,1) \simeq \partial M \times (0,1)$ and so also the homotopy equivalence $M \cup (\partial M \times [0,1)) \simeq M \cup (\partial M \times (0,1))$.
Hence, it remains to be shown that $M \cup (\partial M \times (0,1) \simeq M \cup (U - \partial M)$, for which it suffices to find a homotopy equivalence $\partial M \times (0,1) \simeq U - \partial M$. I'm struggling with this final step. My idea is the following:
Consider the diffeomorphism $f: U \to \partial M \times [0,1)$ given by the collar neighbourhood theorem. If we can show that $\text{Im}(f|_{\partial M}) = \partial M \times \lbrace 0 \rbrace$, then we define $\tilde{f}= f|_{U - \partial M}$. Then $\tilde{f}$ and $\tilde{f^{-1}} = f^{-1}|_{\partial M \times (0,1)}$ are continuous as the restrictions of a continuous functions, and it holds that $\tilde{f} \circ \tilde{f^{-1}} = id_{\partial M \times (0,1)}$ and that $\tilde{f^{-1}} \circ \tilde{f} = id_{U - \partial M}$, hence we obtain the desired homotopy equivalence $\partial M \times (0,1) \simeq U - \partial M$.
My question is whether or not this approach is reasonable, and if so, how I can show that $\text{Im}(f|_{\partial M}) = \partial M \times \lbrace 0 \rbrace$. It seems clear to me that if $\text{Im}(f|_{\partial M}) \neq \partial M \times \lbrace 0 \rbrace$ then there would be contradictions to the bijectivity or continuity of $f$, however I would like to know if there is an easier way to show this before trying to elaborate on this.