Defining the Characteristic Function $ \quad \phi(t) := \mathbb{E} \left[ e^{itx} \right] $ for a random variable with distribution function $F(x)$ in order to show it is uniformly continuous I say
$ |\phi(t+u) - \phi(t)| = \left |\int e^{itx}(e^{iux} - 1) dF(x) \right| \le \\ \int 1 \cdot|e^{iux} -1|dF(x) \to 0 \quad as \quad u\to0 $
Now my question is, does the convergence I state in the last line follow directly, or do I need to be a little carful before I conclude it is true ? (i.e. can I directly use that $|e^{itu} -1| \to 0 ? )$