The Weierstrass M-Test

In the setting of The Real Numbers: Standing Notation and Background, let $D$ be a set, let $S\subseteq D$, let $(g_{k})_{k\in\mathbb{N}}$ be a sequence in the set of functions from $D$ to $\mathbb{R}$, and let $(s_{m})_{m\in\mathbb{N}}$ be its sequence of partial sums. Write $|t|$ for the absolute value of $t\in\mathbb{R}$, and let convergence of a series of real numbers, its sum, and absolute convergence be as defined there.

Let $(M_{k})_{k\in\mathbb{N}}$ be a sequence of real numbers such that $0\le M_{k}$ for every $k\in\mathbb{N}$, such that the series $\sum_{k=1}^{\infty}M_{k}$ converges, and such that

For $m\in\mathbb{N}$ put

the second sum being the $m$-th partial sum of $(M_{k})_{k\in\mathbb{N}}$. Then the following hold.

1. (Absolute convergence at each point)¶ For every $x\in S$ the series $\sum_{k=1}^{\infty}g_{k}(x)$ converges absolutely, and in particular converges.

2. (Uniform tail bound)¶ The real numbers $\tau_{m}$ satisfy $0\le\tau_{m}$ for every $m\in\mathbb{N}$, and the sequence $(\tau_{m})_{m\in\mathbb{N}}$ converges to $0$. If moreover $f:D\to\mathbb{R}$ satisfies $f(x)=\sum_{k=1}^{\infty}g_{k}(x)$ for every $x\in S$, then

3. (Uniform convergence)¶ If $f:D\to\mathbb{R}$ satisfies $f(x)=\sum_{k=1}^{\infty}g_{k}(x)$ for every $x\in S$, then the series $\sum_{k=1}^{\infty}g_{k}$ converges uniformly to $f$ on $S$.