Let $\left(u_{n}(x): n=0,1,2, \ldots\right)$ be a sequence of real-valued functions defined on a subset $E$ of $\mathbb{R}$. Suppose that for all $n$ and all $x \in E$ we have $\left|u_{n}(x)\right| \leqslant M_{n}$, where $\sum_{n=0}^{\infty} M_{n}$ converges. Prove that $\sum_{n=0}^{\infty} u_{n}(x)$ converges uniformly on $E$.

Now let $E=\mathbb{R} \backslash \mathbb{Z}$, and consider the series $\sum_{n=0}^{\infty} u_{n}(x)$, where $u_{0}(x)=1 / x^{2}$ and

$$u_{n}(x)=1 /(x-n)^{2}+1 /(x+n)^{2}$$

for $n>0$. Show that the series converges uniformly on $E_{R}=\{x \in E:|x|<R\}$ for any real number $R$. Deduce that $f(x)=\sum_{n=0}^{\infty} u_{n}(x)$ is a continuous function on $E$. Does the series converge uniformly on $E$ ? Justify your answer.