prove $\frac{e^{iuX_t}} {\mathrm{E}{[e^{iuX_t}]}}$ is martingale

Question

...knowing that $X_t$ has independent increments and is adapted to its natural filtration, $u \in \mathrm{R}$

My problem is in particular how to show this process has finite mean...(can I use the fact that $e^{iux}$ is bounded $\forall x\in\mathrm{R}$?)

EDIT $E[e^{iuX_t}]\neq 0$