Existence of a sequence. While reading about Weierstrass' Theorem and holomorphic functions, I came across a statement that said: "Let $U$ be any connected open subset of $\mathbb{C}$ and let $\{z_j\} \subset U$ be a sequence of points that has no accumulation point in $U$ but that accumulates at every boundary point of $U$."
I was curious as to why such a sequence exists. How would I be able to construct such a sequence?

Construct your sequence in stages indexed by positive integers $N$.  At stage $N$, enumerate those points $(j+ik)/N^2$ for integers $j,k$ with $|j|+|k|\le N^3$ that are within distance $1/N$ of the boundary of $G$.  
EDIT: Oops, not so clear that this will give you points in $G$.  I'll have to be a bit less specific.  At stage $N$, consider $K_N = \partial G \cap \overline{D_N(0)}$ where $\partial G$ is the boundary of $G$ and $D_r(a) = \{z: |z-a|&ltr\}$ .  Since this is compact, it can be covered by finitely many open disks $D_{1/N}(a_k)$, $k=1,\ldots,m$, centred at points $a_k \in K_N$.  Since $a_k \in \partial G$, $G \cap D_{1/N}(a_k) \ne \emptyset$.  So we take a point $z_j \in G \cap D_{1/N}(a_k)$ for $k=1,\ldots,m$.     
