A First Course in Complex Analysis · Problem 7.5.19

Taylor and Laurent Series — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · uniform-convergence

Problem

(a) Suppose

G \subseteq \mathbb{C}

and

f_n : G \to \mathbb{C}

for

n \geq 1

. Suppose

(a_n)

is a sequence in

\mathbb{R}

with

\lim_{n\to\infty} a_n = 0

and, for each

n \geq 1

|f_n(z)| \leq a_n \quad \text{for all } z \in G.

Show that

(f_n)

converges uniformly to the zero function in

G

. (b) Re-prove the statement of Example 7.3.3 using part (a).