Complex Analysis: Theory and Applications · Problem 4.1.15

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · sequence-unique-limit

Problem

Prove that a sequence can have only one limit. Hint: Suppose that there is a sequence

\{z_n\}

such that

z_n \to \zeta_1

and

z_n \to \zeta_2

. Show this assumption implies

\zeta_1 = \zeta_2

by proving that for all

\varepsilon > 0

|\zeta_1 - \zeta_2| < \varepsilon