Lebl-Basic-Analysis-I · Problem 2.3.6

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 such that xn≤ynx_n \leq y_n for all nn. Show lim sup⁡n→∞xn≤lim sup⁡n→∞ynandlim inf⁡n→∞xn≤lim inf⁡n→∞yn.\limsup_{n\to\infty} x_n \leq \limsup_{n\to\infty} y_n \quad \text{and} \quad \liminf_{n\to\infty} x_n \leq \liminf_{n\to\infty} y_n.

Related theory

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