Suppose that the complex series {zn} converges to ζ. Show that {zn} is bounded in two ways.
(a) Write zn=xn+iyn and use the fact that convergent series of real numbers are bounded.
(b) For ε=1, use Definitions 4.1 and 4.2 to show that there is some integer N such that, for n>N, ∣zn∣=∣ζ+(zn−ζ)∣≤∣ζ∣+1. Then set R=max{∣z1∣,∣z2∣,…,∣zN∣,ζ+1}.