This question comes with two parts.
Part (a): Let $\{f_n(x)\}$ be a sequence of nonnegative functions for $x \in S \subseteq \mathbb{R}$ such that $f_1 \geq f_2 \geq \dots \geq 0$, and that $f_n \to 0$ uniformly on $S$. Prove that
$$\sum_{n = 1}^{\infty}(-1)^{n - 1}f_n(x)$$
converges uniformly on $S$.
Part (b): Use part (a) to determine if
$$\sum_{n = 1}^{\infty}\frac{(-1)^{n - 1}}{\sqrt{n(n + x)}}$$
converges uniformly on $(0, \infty)$.
I was able to prove part (a). I am getting stuck on part (b). I think the sequence of functions in part (b) only converges pointwise, but I am struggling with this proof. The proof would be a lot simple if $x = 0$, but it cannot be. Do I do something with infimum's here?
Since the other conditions apply trivially, you just need to check that $f_n(x)=\frac{1}{\sqrt{n(n+x)}}$ converges uniformly to $0$ in $(0,\infty)$. Just note that
$$ \sup_{x >0} \left|\frac{1}{\sqrt{n(n+x)}} \right| \leq \frac{1}{n} \to 0\quad(n \ \to \infty) $$