1D manifold is diffeomorphic to $\mathbb R$ or to $S^1$

Question

In his ODE classic V.I. Arnold considers easy to see (легко видеть) that every one-dimensional (connected and without boundary) differentiable manifold is either diffeomorphic to $\mathbb R$ (if it is not compact) or to $S^1$ (if it is compact).

Is it indeed that easy? Can this be shown using elementary tools?

Answer 1

I think Milnor's "Topology from a Differentiable Viewpoint" (accessible online) has a proof of this in the appendix. The proof is pretty simple, but uses one slightly technical lemma. I would say the Theorem is easy to see since the idea is straightforward, but requires slightly more work to prove.

Here it is. See page 55.