Say {$\phi_n$} be a sequence of characteristic function with some densities {$f_n$} that converges pointwise to $\phi$ , i.e., $\lim_{n\to\infty} \phi_n (t) = \phi(t) \forall t\in \mathbb{R}$ . If $\phi$ is continuous at 0 , how to show $\phi$ is a characteristic function ?
Is it like $\phi$ satisfies basic properties like it is continuous , $\phi(0)=1$ ,$|\phi|\le1$ , so $\phi$ is a characteristic function ? But if that is the case why these basic properties implies $\phi$ being a characteristic function ?
$X_{n}\xrightarrow{d} X\iff \phi_{X_{n}}(t)\xrightarrow{n\to\infty}\phi_{X}(t)$ .
So now suppose you construct $X_{1},X_{2},..$ such that they have their cf as $\phi_{X_{n}}$.
It can be shown that this sequence of rv's is a tight sequence.
Then using Tightness you get a subsequence and a random variable $X$ such that $X_{n_{k}}\xrightarrow{d} X$.
So $\phi=\phi_{X}$ ( using pointwise convergence).
Which in particular implies that $\phi$ is a cf as it agrees with $\phi_{X}$ which is a cf .
For the proof of Tightness you can refer to Rick Durrett as suggested in the comments on page 132-133. Here is a link for the pdf