= Solution
The <Lévy continuity theorem> states that if <characteristic functions> $\varphi_n$ of <probability distributions> $\mu_n$ converge pointwise on $\mathbb R$ to $\varphi$, and $\varphi$ is continuous at zero, then $\varphi$ is the <characteristic function> of a unique <probability distribution> $\mu$ and
$$
\boxed{\mu_n\Rightarrow\mu.}
$$
Here the arrow denotes <weak convergence of probability measures>, equivalently <convergence in distribution> of associated <random variables>. Conversely, <weak convergence of probability measures> implies pointwise convergence of their <characteristic functions> to the <characteristic function> of the limit. The continuity-at-zero condition is essential to ensure that the pointwise limit is a <characteristic function>.
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