Let be the real and imaginary parts of the complex number z. The absolute value |z| is of complex terms αk is said to converge absolutely when the real-valued sums both converge absolutely. Prove...

Let

be the real and imaginary parts of the complex number z. The absolute value |z| is

of complex terms α_k
is said to converge absolutely when the real-valued sums

both converge absolutely. Prove that

converges absolutely if and only if there is a bounding constant B such that