calculus - Imaginary part of $f_m$ tends to zero

Does anybody have an idea how to show that for $|x|< \pi$ the imaginary part of the following sequence of functions $f_m$ tends to zero for $m \rightarrow \infty.$

$$f_m(x):=\left( \frac{e^{-ix}-1}{-ix} \right)^m \left( \sum_{l \in \mathbb{Z}} \frac{\left|e^{-ix}-1 \right|^{2m}}{|x+ 2 \pi l |^{2m}} \right)^{-\frac{1}{2}}$$

So I want to show that $Im(f_m)|_{(-\pi,\pi)} \rightarrow 0.$ Unfortunately, I don't get anywhere because I don't know how to estimate the imaginary part of this product there.

The question is part of a longer proof in analysis.