The Weierstrass M-test gives absolute pointwise convergence and uniform convergence of a function series

Given: Functions $f_k:X\to\mathbb{R}$ and nonnegative reals $M_k$ with $|f_k(x)|\le M_k$ for all $k,x$, such that $\sum M_k$ converges.

If $0\le a_k\le b_k$ eventually and $\sum b_k$ converges, then $\sum a_k$ converges (If $0 \le a_k \le b_k$ eventually, convergence of $\sum b_k$ gives convergence of $\sum a_k$, and divergence of $\sum a_k$ gives divergence of $\sum b_k$).

A convergent real series has uniformly small scalar tails: for every real $\varepsilon>0$ there is $N$ such that $\left|\sum_{k=m+1}^{n}a_k\right|<\varepsilon$ whenever $n>m\ge N$ (A series converges iff for every $\varepsilon > 0$ there is $N$ with $|a_{m+1} + \dots + a_n| < \varepsilon$ for all $n > m \ge N$).

Repeated triangle inequalities for finite sums give $\left|\sum_{k=p}^{q}u_k\right|\le\sum_{k=p}^{q}|u_k|$, and finite sums preserve termwise inequalities (The triangle inequality, Basic properties of the absolute value, Finite sums and finite products, by recursion, Laws of finite sums and finite products).

A function series converges uniformly exactly when its tails are uniformly small (A series of real-valued functions converges uniformly if and only if its tails are uniformly small, A series of real-valued functions and its pointwise and uniform convergence through its partial sums).