Dominated convergence theorem on $e^{ix}$
I am considering first:
$$\lim_{n \to 0} \int_{0}^{\pi} e^{ine^{ix}} dx$$
To bring the limit inside I need to apply the dominated convergence theorem. Keep in mind **I have no knowledge of measure theory...**
I was thinking if:
$$e^{ix} \le x$$ we can do something.
But we cannot compare a complex function with a real function. We **could** set a bound, $x=\pi$ such that:
$$ e^{ine^{ix}} \le e^{-in}$$
But we cannot use the dominated convergence theorem, because we a function of $x$, $g(x)$.
I cannot prove uniform continuity either. What to do?
**The book says we CAN take the limit inside**