Lebl-Basic-Analysis-I · Problem 2.3.7

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. a) Show that {xn+yn}n=1∞\{x_n + y_n\}_{n=1}^\infty is bounded. b) Show that (lim inf⁡n→∞xn)+(lim inf⁡n→∞yn)≤lim inf⁡n→∞(xn+yn).\left(\liminf_{n\to\infty} x_n\right) + \left(\liminf_{n\to\infty} y_n\right) \leq \liminf_{n\to\infty}(x_n + y_n). Hint: One proof is to find a subsequence {xnm+ynm}m=1∞\{x_{n_m} + y_{n_m}\}_{m=1}^\infty of {xn+yn}n=1∞\{x_n + y_n\}_{n=1}^\infty that converges. Then find a subsequence {xnmi}i=1∞\{x_{n_{m_i}}\}_{i=1}^\infty of {xnm}m=1∞\{x_{n_m}\}_{m=1}^\infty that converges. c) Find an explicit {xn}n=1∞\{x_n\}_{n=1}^\infty and {yn}n=1∞\{y_n\}_{n=1}^\infty such that (lim inf⁡n→∞xn)+(lim inf⁡n→∞yn)<lim inf⁡n→∞(xn+yn).\left(\liminf_{n\to\infty} x_n\right) + \left(\liminf_{n\to\infty} y_n\right) < \liminf_{n\to\infty}(x_n + y_n). Hint: Look for examples that do not have a limit.

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