[proofplan] The argument uses the triangle inequality to show the partial sums of $\sum g_n(x)$ are Cauchy, given that the partial sums of $\sum |g_n(x)|$ are Cauchy. Completeness of $\mathbb{R}$ then delivers convergence. [/proofplan] [step:Show the partial sums are Cauchy using absolute convergence and the triangle inequality] Since $\sum_{n=1}^\infty |g_n(x)|$ converges, its partial sums are Cauchy: for every $\varepsilon > 0$, there exists $K \in \mathbb{N}$ such that for all $N > M > K$, \begin{align*} \sum_{j=M+1}^{N} |g_j(x)| &< \varepsilon. \end{align*} 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)| < \varepsilon. \end{align*} Therefore the partial sums $S_N(x) = \sum_{j=1}^N g_j(x)$ form a [Cauchy sequence](/page/Cauchy%20Sequence) in $\mathbb{R}$. [guided] We want to show that $\sum g_n(x)$ converges at the fixed point $x$. The hypothesis is that $\sum |g_n(x)|$ converges, meaning its partial sums are Cauchy. We transfer this to the original [series](/page/Series) via the triangle inequality. For $N > M$, the tail of the partial sums satisfies \begin{align*} \left|S_N(x) - S_M(x)\right| &= \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 the tail of the absolutely convergent series $\sum |g_j(x)|$. Since that series converges, its tails can be made arbitrarily small: for any $\varepsilon > 0$, there exists $K$ such that $\sum_{j=M+1}^{N} |g_j(x)| < \varepsilon$ for all $N > M > K$. This makes $(S_N(x))$ a Cauchy sequence in $\mathbb{R}$. [/guided] [/step] [step:Conclude convergence from completeness of $\mathbb{R}$] The partial sums $S_N(x) = \sum_{j=1}^N g_j(x)$ form a [Cauchy sequence](/page/Cauchy%20Sequence) in $\mathbb{R}$. By completeness of $\mathbb{R}$, $(S_N(x))$ converges, so $\sum_{n=1}^\infty g_n(x)$ converges. [/step]