Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics II / 3 / a / ii / Solution

Let $T_N={\sum }_{n=1}^N|a_n|$ and $S_N={\sum }_{n=1}^N{a}_n$. If $(T_N)$ converges, it is a Cauchy sequence. Thus, for every $\epsilon >0$, some $N$ makeswhere the triangle inequality gives the bound. Therefore $(S_N)$ is a Cauchy sequence in the complex numbers. These form a complete metric space, so $(S_N)$ has a finite limit of a sequence. Equivalently, apply the Cauchy criterion for series. Hence absolute convergence implies convergence.