I'm trying to prove the following that if a random variable converges in probability, then the characteristic function converges pointwise. This is: $$ X_n \overset{p}{\to} X \quad \implies \quad \lim_{n\to\infty}\psi_{X_n}(t) = \psi_X(t) \ \forall t $$ I should be able to prove it using the theorem below, with no measure theory and no use of almost sure convergence. Those are out of the scope of my study.
If $X_n \overset{p}{\to} X$ and for all $n$ $|X_n|\leq Y$, where $Y\in L^p$, then $|X|\in L^p$ and $ X_n \overset{L_p}{\to} X$
My problem is that I don't see any connection between the converge in probability or $L^p$, that as far as I can tell relate to moments, and the characteristic function which is an expectation of infinite complex momenta. Any hints on the connection or similar proofs would be preferred.