The fundamental theorem of algebra
Theorem: Any non-constant polynomial over $\mathbb{C}$ has a zero.
Proof:
Let $f(z)$ be a polynomial. Assume by contradiction that $f(z)$ does not have a zero. Define a function $g(z) = \frac{1}{f(z)}$. Then $g(z)$ is an entire function, and satisfies $\lim_{z\to\infty} g(z) = 0$. Since it is also bounded in any compact set, it follows that $g(z)$ is a bounded entire function. By Liouville's theorem, $g(z)$ is constant, and therefore $f(z)$ is a constant polynomial.