A First Course in Complex Analysis · Problem 7.5.20

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

Mathematics — Complex Analysis · uniform-convergence

Problem

(a) Suppose

G \subseteq \mathbb{C}

f_n : G \to \mathbb{C}

for

n \geq 1

, and

(f_n)

converges uniformly to the zero function in

G

. Show that, if

(z_n)

is any sequence in

G

, then

\lim_{n\to\infty} f_n(z_n) = 0.

(b) Apply (a) to the function sequence given in Example 7.3.2, together with the sequence

(z_n = e^{-\frac{1}{n}})

, to prove that the convergence given in Example 7.3.2 is not uniform.