From Levy's continuity theorem it is known that pointwise convergence of characteristic functions is equivalent to the convergence in law of corresponding random variable. However, I have to show that if $X_n \sim\mathrm{Unif} [-n,n]$, then the characteristic functions of the sequence converge pointwise, but $(X_n)_{n \in \mathbb{N}}$ does not converge in law.
I have calculated the characteristic function which is $\frac{\sin(tn)}{tn}$ and which converges to 0. Still I do not know yet how to show, that the sequence of r.v. does not converge in law and more important, why this example does not contradict the stated theorem ?