Let $Y$ be a non-singular curve which is birational to but not isomorphic to $\mathbb{P}^1$. How can one see that this curve is isomorphic to an open subset of the affine line? This is Hartshorne problem I.6.1.
Thoughts so far: By the definition of birational equivalence, there is an isomorphism $\varphi: U \to V$ of open subsets of $Y$ and $\mathbb{P}^1$, respectively. Since $Y$ has the cofinite topology, it should be possible to use Hartshorne's Proposition I.6.8 to extend $\varphi$ to $Y$. However, I can't see why the map so defined should be an isomorphism.