Lebl-Basic-Analysis-I · Problem 2.3.13

Sequences and Series

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

Problem

(Easy) Suppose {xn}n=1∞\{x_n\}_{n=1}^\infty is such that lim inf⁡n→∞xn=−∞\displaystyle\liminf_{n\to\infty} x_n = -\infty, lim sup⁡n→∞xn=∞\displaystyle\limsup_{n\to\infty} x_n = \infty. a) Show that {xn}n=1∞\{x_n\}_{n=1}^\infty is not convergent, and also that neither lim⁡n→∞xn=∞\lim_{n\to\infty} x_n = \infty nor lim⁡n→∞xn=−∞\lim_{n\to\infty} x_n = -\infty is true. b) Find an example of such a sequence.

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