Lebl-Basic-Analysis-I · Problem 2.3.14

Sequences and Series

Mathematics · limit-superior-limit-inferior-bolzano-weierstrass

Problem

Let {xn}n=1∞\{x_n\}_{n=1}^\infty be a sequence. a) Show that lim⁡n→∞xn=∞\lim_{n\to\infty} x_n = \infty if and only if lim inf⁡n→∞xn=∞\displaystyle\liminf_{n\to\infty} x_n = \infty. b) Then show that lim⁡n→∞xn=−∞\lim_{n\to\infty} x_n = -\infty if and only if lim sup⁡n→∞xn=−∞\displaystyle\limsup_{n\to\infty} x_n = -\infty. c) If {xn}n=1∞\{x_n\}_{n=1}^\infty is monotone increasing, show that either lim⁡n→∞xn\lim_{n\to\infty} x_n exists and is finite or lim⁡n→∞xn=∞\lim_{n\to\infty} x_n = \infty. In either case, lim⁡n→∞xn=sup⁡{xn:n∈N}\lim_{n\to\infty} x_n = \sup\{x_n : n \in \mathbb{N}\}.

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