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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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.
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