Uniform convergence of series with nonnegative terms implies uniform convergence of terms to zero

Question

If I tell you that $S_n(x) = \sum^n_{k=0} \lvert f_k(x)\rvert $ converges uniformly, does it follow that $\lvert f_k(x)\rvert$ converge uniformly to zero?

Assume that $f_k$ are real-valued functions defined on the reals.

Answer 1

$S_n$ converges uniformly to some function $S$ iff for every $\epsilon > 0$, there exists an $N(\epsilon)\in \mathbb{N}$ such that for all $n,m\geq N(\epsilon)$ and $x\in \mathbb{R}$ we have $$\lvert S_n(x) - S_m(x)| < \epsilon$$ (This follows directly from the triangle inequality). Setting $n=m+1$, we obtain $$\lvert S_{m+1}(x) - S_m(x)\rvert = \lvert f_{m+1}(x)\rvert < \epsilon\quad \forall x\in \mathbb{R}.$$ This is essentially the same argument as in the proof of the fact that the convergence of $\sum_{n=1}^{\infty} a_n$ implies $a_n\to 0, n\to \infty.$