uniform convergence of $\sum\limits_{n=1}^{\infty}f_n(x)$ implies $\lim\limits_{n\to\infty}f_n(x)=0$ uniformally

Question

The question is:

Let $\displaystyle \sum_{n=1}^{\infty}f_n(x)$ a series converges uniformly on $I$ to $S(x)$. Prove that $\lim\limits_{n\to\infty}f_n(x)=0$.

My try: $\forall N\in\mathbb N$, we have $f_N(x)=\sum\limits_{n=1}^{N}f_n(x)-\sum\limits_{n=1}^{N-1}f_n(x)$, so the statement is proved if I prove that

$$f_n(x),g_n(x)\to f(x),g(x)\text{ uniformly on }J\implies\\ f_n(x)+g_n(x)\to f(x)+g(x) \text{ uniformly on }J.$$ On the one hand: $$\begin{align*} \lim_{n\to\infty}\sup_{x\in J}|f_n(x)-f(x)+g_n(x)-g(x)|\leq&\quad\text{(triangle inequality) }\\ \limsup\limits_{x\in J}{|f_n(x)-f(x)|+|g_n(x)-g(x)|}=0+0=0. \end{align*}$$

Now I'm in trouble proving that $\limsup\limits_{x\in J}{|f_n(x)+g_n(x)-f(x)-g(x)|}\geq 0$. How can I do that ?

Answer 1

We know:
$\forall x\in J:|f_n(x)+g_n(x)-f(x)-g(x)|\ge0 $
then:
$\limsup\limits_{x\in J}{|f_n(x)+g_n(x)-f(x)-g(x)|}\ge0$

Answer 2

It easier to work with the remainder $R_n(x) = \sum_{k=n}^\infty f_k(x)$. By hypothesis, we know that $$ \lim_{n\to\infty} \sup_{x} |R_n(x)| = 0. $$

Since $f_n(x) = R_n(x) - R_{n+1}(x)$, we deduce $$ \sup_x |f_n(x)| \leq \sup_x |R_n(x)| + \sup_x |R_{n+1}(x)| \xrightarrow[n\to\infty]{} 0. $$