A First Course in Complex Analysis · Problem 7.5.5

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

Mathematics — Complex Analysis · sequence-limit-properties

Problem

Prove the following: (a) lim⁡n→∞an=a⟹lim⁡n→∞∣an∣=∣a∣\lim_{n\to\infty} a_n = a \Longrightarrow \lim_{n\to\infty}|a_n| = |a|. (b) lim⁡n→∞an=0⟺lim⁡n→∞∣an∣=0\lim_{n\to\infty} a_n = 0 \Longleftrightarrow \lim_{n\to\infty}|a_n| = 0.

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