[proofplan] We show the partial sums $S_N = \sum_{j=1}^N g_j$ form a uniformly [Cauchy sequence](/page/Cauchy%20Sequence) by bounding the tail $|S_N(x) - S_M(x)|$ with the tail of the convergent numerical [series](/page/Series) $\sum M_n$. The [Cauchy Criterion for Uniform Convergence](/theorems/257) then delivers [uniform convergence](/page/Uniform%20Convergence). Absolute convergence follows from the pointwise comparison test. [/proofplan] [step:Bound the partial-sum tails uniformly using the convergent series $\sum M_n$] Define the partial sums $S_N: E \to \mathbb{R}$ by $S_N(x) = \sum_{j=1}^N g_j(x)$. Fix $\varepsilon > 0$. Since $\sum_{n=1}^\infty M_n$ converges, its partial sums form a Cauchy [sequence](/page/Sequence) in $\mathbb{R}$: there exists $K \in \mathbb{N}$ such that $\sum_{n=M+1}^N M_n < \varepsilon$ for all $N > M \geq K$. For any $x \in E$ and any $N > M \geq K$, the triangle inequality gives \begin{align*} |S_N(x) - S_M(x)| &= \left| \sum_{n=M+1}^N g_n(x) \right| \leq \sum_{n=M+1}^N |g_n(x)| \leq \sum_{n=M+1}^N M_n < \varepsilon. \end{align*} Since this bound holds for all $x \in E$, we have $\sup_{x \in E} |S_N(x) - S_M(x)| < \varepsilon$, so the partial sums $(S_N)$ are uniformly Cauchy on $E$. [/step] [step:Apply the Cauchy criterion to obtain uniform convergence] By the previous step, the [sequence](/page/Sequence) of partial sums $(S_N)$ is uniformly Cauchy on $E$. By the [Cauchy Criterion for Uniform Convergence](/theorems/257), $(S_N)$ converges uniformly on $E$. By definition, this means $\sum_{n=1}^\infty g_n$ converges uniformly on $E$. [/step] [step:Verify absolute convergence pointwise via the comparison test] Fix $x \in E$. For each $n \in \mathbb{N}$, $|g_n(x)| \leq M_n$. Since $\sum_{n=1}^\infty M_n$ converges and $|g_n(x)| \leq M_n$ for all $n$, the comparison test for non-negative numerical [series](/page/Series) gives $\sum_{n=1}^\infty |g_n(x)| \leq \sum_{n=1}^\infty M_n < \infty$. Therefore $\sum_{n=1}^\infty g_n(x)$ converges absolutely at each $x \in E$. [/step]