1-manifold is orientable I am trying to classify all compact 1-manifolds. I believe I can do it once I can show every 1-manifold is orientable. I have tried to show prove this a bunch of ways, but I can't get anywhere.
Please help,
Note, I am NOT assuming that I already know the only such manifolds are [0,1] or $S^1$. This is my end goal.  

If you've already classified orientable $1$-manifolds, then you know that the only connected ones (without boundary) are $\mathbb R$ and $\mathbb S^1$. Now suppose $M$ is a connected, nonorientable $1$-manifold, and let $\pi\colon \widetilde M\to M$ be its universal covering.  Then $\widetilde M$ is orientable and simply connected, and therefore homeomorphic to $\mathbb R$, and $M$ is homeomorphic to a quotient of $\mathbb R$ by a free group action that does not preserve orientation.  The last step is to show that every orientation-reversing homeomorphism $f\colon \mathbb R\to \mathbb R$ has a fixed point, which yields a contradiction.  Thus every $1$-manifold is orientable.
For a $1$-manifold with (nonempty) boundary, you can apply the above argument to the double of $M$ (the quotient of two disjoint copies of $M$ obtained by identifying each boundary point in one copy with the corresponding boundary point in the other).
