Absolutely convergent

Proof that any absolutely convergent series of complex numbers is convergent

Suppose that ${\textstyle \sum \left|a_{k}\right|,a_{k}\in \mathbb {C} }$ is convergent. Then equivalently, ${\textstyle \sum \left[\operatorname {Re} \left(a_{k}\right)^{2}+\operatorname {Im} \left(a_{k}\right)^{2}\right]^{1/2}}$ is convergent, which implies that ${\textstyle \sum \left|\operatorname {Re} \left(a_{k}\right)\right|}$ and ${\textstyle \sum \left|\operatorname {Im} \left(a_{k}\right)\right|}$ converge by termwise comparison of non-negative terms. It suffices to show that the convergence of these series implies the convergence of ${\textstyle \sum \operatorname {Re} \left(a_{k}\right)}$ and ${\textstyle \sum \operatorname {Im} \left(a_{k}\right),}$ for then, the convergence of ${\textstyle \sum a_{k}=\sum \operatorname {Re} \left(a_{k}\right)+i\sum \operatorname {Im} \left(a_{k}\right)}$ would follow, by the definition of the convergence of complex-valued series.