Lévy's Continuity Theorem

Every probability distribution has a characteristic function — essentially its Fourier transform — which encodes the same information as the distribution itself but is often much easier to manipulate algebraically, especially for sums of random variables. Lévy's continuity theorem is the bridge that lets you go back the other way: if you can show a sequence of characteristic functions converges, pointwise, to some limiting function that is continuous at zero, then the theorem guarantees the underlying distributions themselves converge in distribution to whatever distribution has that limiting function as its characteristic function.

This matters because directly proving that a complicated sequence of distributions converges to, say, the normal distribution can be very difficult, while showing that their characteristic functions converge to the normal's characteristic function is often a short algebraic computation. This is exactly the mechanism used to prove the central limit theorem: the characteristic function of a standardized sum of i.i.d. random variables can be shown, via a Taylor expansion, to converge to $e^{-t^2/2}$ — the characteristic function of the standard normal — and Lévy's theorem then hands you the conclusion that the sums themselves converge in distribution to normal.

In the CLT proof, once you've shown the standardized sum's characteristic function $\varphi_n(t) \to e^{-t^2/2}$ for every fixed $t$, Lévy's continuity theorem is the single step that upgrades "the characteristic functions converge" into "the distributions converge to $N(0,1)$" — without it, the characteristic-function computation alone would prove nothing about the distributions people actually care about.