= Solution
{c}
Let $T_N=\sum_{n=1}^N|a_n|$ and $S_N=\sum_{n=1}^Na_n$. If $(T_N)$ converges, it is a <Cauchy sequence>. Thus, for every $\varepsilon>0$, some $N$ makes
$$
\left|S_q-S_p\right|=\left|\sum_{n=p+1}^qa_n\right|\leq\sum_{n=p+1}^q|a_n|=T_q-T_p<\varepsilon
\qquad(q>p\geq N),
$$
where 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 \b[<absolute convergence> implies convergence.]
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