Paul Lévy continuity theorem

Let $F_1,F_2,\mathrm{\dots }$ be distribution functions with characteristic functions${\phi }_1,{\phi }_2,\mathrm{\dots }$, respectively. If ${\phi }_n$ converges pointwiseto a limit $\phi$, and if $\phi (t)$ is continuous at $t=0$, thenthere exists a distribution function $F$ such that $F_n\to F$ weakly (http://planetmath.org/ConvergenceInDistribution), and the characteristic function associated to $F$ is $\phi$.

Remark. The reciprocal of this theorem is a corollary to the Helly-Bray theorem; hence $F_n\to F$ weakly if and only if ${\phi }_n\to \phi$ pointwise; but this theorem says something stronger than the sufficiency of that : it says that the limit of a sequence of characteristic functions is a characteristic function whenever it is continuous at 0.