I have this problem: If $(a_n)_n$ is a sequence of real numbers and $(e^{i u a_n})_n$ converges to finite limit $g(u)$ for all $u\in I\subset\mathbb{R}$. Show that $(a_n)_n$ converges.
I tried to prove using Levy's continuity theorem, but the problem is the continuity of the limit of $(e^{i u a_n})_n$ at $u=0$. Someone can give me any hint for this part of the continuity ?.
This is an exercise (Chapter 8 Ex4) of the Book “real analysis and probability ” by Robert Ash.
By dominated convergence theorem, the following convergence holds for each $b<c$: $$ \lim_{n\to\infty}\int_b^c e^{iua_n}du=\int_b^cg(u)du. $$ Fix $b,c$ such that $b<c$ and $\int_b^cg(u)du\neq 0$ (such $b,c$ exist, otherwise, $g$ would have an integral that vanishes over every interval hence would be identically zero, which is not possible, since $\lvert g(u)\rvert=1$). Then express $ a_n\int_b^ce^{ia_nu}du $ as a constant times $e^{ica_n}-e^{icb_n}$. For $n$ large enough, you can divide by $\int_b^ce^{ia_nu}du$ and derive the wanted convergence.