Let $\left(f_{n}\right)_{n \geqslant 1}$ be a sequence of continuous complex-valued functions defined on a set $E \subseteq \mathbb{C}$, and converging uniformly on $E$ to a function $f$. Prove that $f$ is continuous on $E$.

State the Weierstrass $M$-test for uniform convergence of a series $\sum_{n=1}^{\infty} u_{n}(z)$ of complex-valued functions on a set $E$.

Now let $f(z)=\sum_{n=1}^{\infty} u_{n}(z)$, where

$$u_{n}(z)=n^{-2} \sec (\pi z / 2 n) .$$

Prove carefully that $f$ is continuous on $\mathbb{C} \backslash \mathbb{Z}$.

[You may assume the inequality $|\cos z| \geqslant|\cos (\operatorname{Re} z)| \cdot]$