[proofplan] The triangle inequality bounds the tail of $\sum g_n$ by the tail of $\sum |g_n|$. Since absolute uniform convergence means the partial sums of $\sum |g_n|$ are uniformly Cauchy, the same bound forces the partial sums of $\sum g_n$ to be uniformly Cauchy, and hence uniformly convergent. [/proofplan] [step:Establish the uniform Cauchy condition for $\sum g_n$ via the triangle inequality] Since $\sum_{n=1}^\infty |g_n|$ converges uniformly on $E$, its partial sums are uniformly Cauchy: for every $\varepsilon > 0$, there exists $K \in \mathbb{N}$ such that for all $N > M \geq K$, \begin{align*} \sup_{x \in E} \sum_{j=M+1}^{N} |g_j(x)| < \varepsilon. \end{align*} For any $x \in E$, the triangle inequality gives \begin{align*} \left|\sum_{j=M+1}^{N} g_j(x)\right| \leq \sum_{j=M+1}^{N} |g_j(x)| \leq \sup_{x \in E} \sum_{j=M+1}^{N} |g_j(x)| < \varepsilon. \end{align*} Taking the supremum over $x \in E$ on the left-hand side preserves the bound, so \begin{align*} \sup_{x \in E} \left|\sum_{j=M+1}^{N} g_j(x)\right| < \varepsilon \quad \text{for all } N > M \geq K. \end{align*} [guided] We want to show that if the series of absolute values $\sum |g_n|$ converges uniformly, then the original series $\sum g_n$ converges uniformly. The natural tool is the triangle inequality, which relates a sum to the sum of absolute values. Since $\sum_{n=1}^\infty |g_n|$ converges uniformly on $E$, its sequence of partial sums is uniformly Cauchy: for every $\varepsilon > 0$, there exists $K \in \mathbb{N}$ such that for all $N > M \geq K$, \begin{align*} \sup_{x \in E} \sum_{j=M+1}^{N} |g_j(x)| < \varepsilon. \end{align*} Now fix any $x \in E$ and any $N > M \geq K$. The triangle inequality applied to the finite sum gives \begin{align*} \left|\sum_{j=M+1}^{N} g_j(x)\right| \leq \sum_{j=M+1}^{N} |g_j(x)|. \end{align*} The right-hand side is bounded by its supremum over $E$, which is less than $\varepsilon$. Since this bound holds for every $x \in E$, we may take the supremum on the left-hand side: \begin{align*} \sup_{x \in E} \left|\sum_{j=M+1}^{N} g_j(x)\right| \leq \sup_{x \in E} \sum_{j=M+1}^{N} |g_j(x)| < \varepsilon. \end{align*} This is precisely the statement that the partial sums of $\sum g_n$ are uniformly Cauchy on $E$. [/guided] [/step] [step:Conclude uniform convergence from the uniform Cauchy condition] The partial sums of $\sum g_n$ are uniformly Cauchy on $E$. Since $\mathbb{R}$ is complete, the pointwise limit $\sum_{n=1}^\infty g_n(x)$ exists for each $x \in E$, and the uniform Cauchy condition implies that the convergence is uniform. Therefore $\sum_{n=1}^\infty g_n$ converges uniformly on $E$. [/step]