I am trying to prove this particular case of the collar neighbourhood theorem. I am following the paper http://www.math.toronto.edu/vtk/1300Fall2015/lecture-nov2.pdf, but the last step is glossed over and labeled as 'easy'.
Let $B$ be the compact boundary of a manifold $M$. Then there exists a neighbourhood $U$ of $B$ in $M$ diffeomorphic to $B \times [0, 1)$.
The proof starts from an inward-pointing vector field $V$ on the boundary, and tries to prove that its flow $F$ (sufficiently restricted) is a diffeomorphism; its inverse will then be the required map.
I have already arrived at the part where I have a finite cover $B_1, \ldots, B_n$ of $B$ so that the flow $F$ is a diffeomorphism from the sets $B_i \times \epsilon_i$ to their images $M_i$ in $M$. So the flow $F$ is a $\textit{local}$ diffeomorphism from $B \times \epsilon$ to its image in $M$, where $\epsilon$ is the minimum of the $\epsilon_i$.
The author of the quoted paper then says:
Using compactness of $∂M$ and arguing by contradiction it’s easy to see that there exists $0 < ε_1 < ε$ such that the flow is injective on $B \times [0, \epsilon_1)$.
I am however at a loss how to prove this exactly. Any help would be appreciated.
ATTEMPT AT PROOF:
Suppose no such $\epsilon_1$ exists. Then we can construct a sequence $b_n$ in $B$, so that $F(b_n, t_n) \in M$ with $t_n < \frac{1}{n}$ is equal to $F(b_n, t_n) = F(c_n, \tau_n)$ for some $c_n \neq b_n$, $\tau_n < \frac{1}{n}$. Said sequence has a limit point $b \in B$, as $B$ is compact. However, $F$ is locally diffeomorphic, so there exists a neighbourhood $V_b$ around $b$ on which $F$ is properly diffeomorphic.
Now magic happens, and we arrive at a contradiction.