Basic Analysis I — Real Analysis · Problem 2.3.17

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · limit-superior-limit-inferior

Problem

Suppose {xn}n=1∞\{x_n\}_{n=1}^\infty is a sequence, and an:=sup⁡{xk:k≥n}a_n := \sup\{x_k : k \geq n\} and bn:=inf⁡{xk:k≥n}b_n := \inf\{x_k : k \geq n\} as before. a) Prove that if aℓ=∞a_\ell = \infty for some ℓ∈N\ell \in \mathbb{N}, then lim sup⁡n→∞xn=∞\limsup_{n\to\infty} x_n = \infty. b) Prove that if bℓ=−∞b_\ell = -\infty for some ℓ∈N\ell \in \mathbb{N}, then lim inf⁡n→∞xn=−∞\liminf_{n\to\infty} x_n = -\infty.

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