Proper Holomorphic Embeddings of Finitely and Some Infinitely Connected Subsets of $\CC$ Into $\CC^2$

We show that any finitely connected domain $U\subset\CC$ can be properly embedded into $\CC^2$. For some sequences $\{p_j\}\subset U$, $U\setminus\{p_j\}$ can also be properly embedded into $\CC^2$.