Lebl-Basic-Analysis-I · Problem 2.3.8

Sequences and Series

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

Problem

Let {xn}n=1∞\{x_n\}_{n=1}^\infty and {yn}n=1∞\{y_n\}_{n=1}^\infty be bounded sequences (by the previous exercise, {xn+yn}n=1∞\{x_n + y_n\}_{n=1}^\infty is bounded). a) Show that (lim sup⁡n→∞xn)+(lim sup⁡n→∞yn)≥lim sup⁡n→∞(xn+yn).\left(\limsup_{n\to\infty} x_n\right) + \left(\limsup_{n\to\infty} y_n\right) \geq \limsup_{n\to\infty}(x_n + y_n). Hint: See previous exercise. b) Find an explicit {xn}n=1∞\{x_n\}_{n=1}^\infty and {yn}n=1∞\{y_n\}_{n=1}^\infty such that (lim sup⁡n→∞xn)+(lim sup⁡n→∞yn)>lim sup⁡n→∞(xn+yn).\left(\limsup_{n\to\infty} x_n\right) + \left(\limsup_{n\to\infty} y_n\right) > \limsup_{n\to\infty}(x_n + y_n). Hint: See previous exercise.

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