Chapter 2.6, Problem 1P

Prove that an absolutely convergent series of complex numbers converges. This means to prove that ${\displaystyle \sum (a_n+i{b}_n)}$ converges (a_n and b_n real) if ${\displaystyle \sum \sqrt{a^2{}_n+b^2{}_n}}$ converges. Hint: Convergence of ${\displaystyle \sum (a_n+i{b}_n)}$ means that ${\displaystyle \sum a_n}$ and ${\displaystyle \sum b_n}$ both converge. Compare ${\displaystyle \sum |a_n}|$ and ${\displaystyle \sum |b_n}|$ with ${\displaystyle \sum \sqrt{a^2{}_n+b^2{}_n}}$ and use Problem 7.9 of Chapter 1.