Problems: The inferior limit of a bounded sequence (an) ∞ n=1 is lim infn an = supn inf m≥n an Show that the inferior limit L = lim infn an has the following properties: (a) For every ε > 0, there exists an n ∈ N, such that am > L − ε for m ≥ n. (b) For every ε > 0, there are infinitely many n ∈ N with an < L + ε.

Problems: The inferior limit of a bounded sequence (an) ∞ n=1 is lim infn an = supn inf m≥n an Show that the inferior limit L = lim infn an has the following properties: (a) For every ε > 0, there exists an n ∈ N, such that am > L − ε for m ≥ n. (b) For every ε > 0, there are infinitely many n ∈ N with an < L + ε.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 72E

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Question

Problems: The inferior limit of a bounded sequence (an)
∞
n=1 is
lim inf_n a_n = sup_n inf _m≥n a_n
Show that the inferior limit L = lim inf_n a_n has the following properties:
(a) For every ε > 0, there exists an n ∈ N, such that am > L − ε
for m ≥ n.
(b) For every ε > 0, there are infinitely many n ∈ N with an < L + ε.