I am trying to figure out the following problem from Q.Liu's algebraic geometry in chapter 4.
Let $U$ be an integral affine algebraic curve over $k$.
(a) show that there exists a proper curve $ \hat{U}$ over $k$ such that there is an open immersion of $U$ into $\hat{U}$ and the points in $\hat{U}-U$ are normal in $\hat{U}$.
(b) any morphism $U\to X$ where $X$ is a proper variety factors uniquely through $\hat{U}$.
(c) show there is a finite morphism $U\to \mathbf{A}^1$, and it extends to a finite morphism $\hat{U}\to\mathbf{P}^1$ such that $U=f^{-1}(\mathbf{A}^1)$
My idea so far is to take the projective closure of $U$. However, I'm not sure how to show that the points we are adding to $U$ in this way are normal.
For the second part, I'm thinking I can imitate the argument that shows a morphism extends uniquely on a normal scheme in this setting. Perhaps the normality of the added points will be enough to extend.
For the last part, I know how to build the morphism to affine space, but I don't know how to show that the extra points have to go the point at infinity.
Any insight into this problem will be greatly appreciated.