The following result, known as the Continuity Theorem is an important step towards a proof of the Central Limit Theorem.
Suppose \(\{ F_{n}(\cdot), n\geq 1 \}\) is a sequence of cumulative distribution functions with the corresponding characteristic functions \(\{ \varphi_{n}(\cdot),n\geq 1 \}\). Result 1: If \(F_{n}\stackrel{d}{\to}F\) for some CDF \(F\) with characteristic function \(\varphi\), then \[ \varphi(t)=\lim_{ n \to \infty } \varphi_{n}(t) \] uniformly in every finite interval. Moreover, \(\{ \varphi_{n},n\geq 1 \}\) is equicontinuous on \(\mathbb{R}\). Result 2: Conversely, if \(\lim_{ n \to \infty }\varphi_{n}(t)=\varphi(t)\) exists for every \(t \in \mathbb{R}\) and \(\varphi(\cdot)\) is continuous at \(t=0\), then \(\varphi(\cdot)\) is a characteristic function of some distribution function \(F(\cdot)\) and \(F_{n}\stackrel{d}{\to}F\).
Proof: