Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ and let $A$ be the Banach algebra of functions that are analytic in $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$ with the uniform norm. I want to show that polynomials of the form $P(z)=\sum_{k=0}^{n}a_{k}z^{k}$ are dense in this algebra.
This is covered by Mergelyan's theorem, which is proved in Rudin's Real and Complex Analysis, but I am wondering if there is a short proof when the hypotheses are simplified to the unit disk. I don't believe it is true that if $f\in A$ and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ on $\mathbb{D}$, then the equality also holds on $\overline{\mathbb{D}}$.