Prove every compact $1$-manifold with boundary always has an even number of boundary points.

Question

There's a claim that Milnor makes in his book Topology from the Differentiable Viewpoint

Every compact $1$-manifold with boundary always has an even number of boundary points.

I'm not quite sure how to prove this, if every compact $1$-manifold with boundary was homeomorphic to a closed interval $[a, b]$ in $\mathbb{R}^1$, then the proof would follow trivially, but I'm not sure if that's a theorem.

How can I go about proving this claim?

Answer 1

There is a full proof of the classification of compact $1$-manifolds with boundary on the end of Milnor's book itself (which states that those are finite unions of circles and closed intervals).