There have been at least two questions on MathOverflow asking about the uniqueness of differentiable structures on $\mathbb{R}^n$ ($n \neq 4$). However, on neither page is an explicit reference or proof of the case I'm interested in: $n=1$.
In the comments of one of them, the following solution is given: "one can put a Riemannian metric on any smooth $\mathbb{R}^1$ and then the exponential map from a given point defines a diffeomorphism from the standard $\mathbb{R}^1$ to the smooth $\mathbb{R}^1$". Unfortunately, I'm not very familiar with either Riemannian metrics or the exponential map.
Does anyone have a proof, or a reference to a proof, of this result that is more elementary? Or is that as simple as it gets?