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, $|z_{n}|\to \infty$ 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 $\mathbb {C}$.