How to prove that this series of functions is uniformly convergent?

Ishraaq Parvez gave you the answer but I'm going to detail. First the theorem about Alternating Series :

Hypothetis :

• $(u_n)$ is alternate, which means either $((-1)^n u_n)_{nin mathbb{N}}$ or $((-1)^{n+1} u_n)_{nin mathbb{N}}$ is positive
• $(|u_n|)$ is decreasing
• $|u_n|rightarrow 0$ when $nrightarrow +infty$

Result :

• $sum u_n$ converges
• if we note $R_n = sumlimits_{i=n+1}{+infty} u_i$, then for all $nin mathbb{N}$, $|R_n|leqslant |u_n|$

Then for your problem : it is clear that for all $x$, $(f_n(x))$ is alternate. We also know that $sum f_n$ uniformly convergent is equivalent to $R_n = sumlimits^{+infty}_{i=n+1} f_i $ uniformly converges to $0$. By the theorem on alternate series, we have first that $sum f_n$ exists, and also that for all $x$, $|R_n|leqslant |f_n(x)|$. So we have that $sup |R_n| leqslant sup |f_n|$. (you will have remarked that the sup are defined since we have convergence)

Since $(f_n)$ uniformly converges to $0$, $sup |f_n| rightarrow 0$. So $sup |R_n| rightarrow 0$. Thus $sum f_n$ uniformly converges.