Weierstrass's theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

Suppose $(z_n)$ is a sequence of (possibly repeating) complex numbers that does not cluster in $\mathbb{C}$: in other words, $\left|z_n\right|\to \mathrm{\infty }$ if the sequence is infinite. Then, there exists an entire function $f:\mathbb{C}\to \mathbb{C}$ such that for any $z\in \mathbb{C}$, the order of zero for $f$ at $z$ equals the number of times $z$ occurs in the sequence $(z_n)$.

Importance

This solves Cousin's multiplicative problem in a particular case over $\mathbb{C}$.