The only 1-manifolds are $\mathbb R$ and $S^1$ I recall having heard somewhere that the only 1-manifolds (second countable, Hausdorff, connected spaces locally homeomorphic to $\mathbb R$) are $\mathbb R$ and $S^1$. Is this true? If so, is there a reasonably elementary proof?

There's a proof outlined in Problems 17-5 and 17-7 of John Lee's "Introduction to Smooth Manifolds" that uses a basic classification of integral curves of vector fields, specifically that a nonconstant maximally defined integral curve is either injective or periodic, which implies (after a small amount of work) that the image of any nonconstant integral curve is diffeomorphic to either $\mathbb{R}$ or $\mathbb{S}^1$.  The problem is finished by showing that any 1-manifold is orientable, and thus admits a nonvanishing global vector field, of which you consider a maximally defined integral curve.
I don't think this is the same proof as given in Guillemin and Pollack or in Milnor, and for my money it's quite a bit simpler than both.
