Bolzano's theorem

If $f$ is continuous on $[a, b]$ and $f(a)$, $f(b)$ have opposite signs, then $f$ has a root in $(a, b)$. It is the guarantee behind the bisection method.

Halve the interval and keep the half where the sign still changes. The nested intervals shrink to a point $c$ (completeness of $\R$), and continuity forces $f(c) = 0$. The proof is the bisection algorithm.