i. For a convergent real sequence sn and a real number a, show that if sn ≥ a for all but finitely many values of n, then limn→∞ sn ≥ a. ii. For each value of a ∈ ℝ, give an example of a convergent sequence sn with sn > a for all n, but where limn→∞ sn = a.

i. For a convergent real sequence sn and a real number a, show that if sn ≥ a for all but finitely many values of n, then limn→∞ sn ≥ a. ii. For each value of a ∈ ℝ, give an example of a convergent sequence sn with sn > a for all n, but where limn→∞ sn = a.

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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i. For a convergent real sequence s_n and a real number a, show that if s_n ≥ a for all but finitely many values of n, then lim_n→∞ s_n ≥ a.
ii. For each value of a ∈ ℝ, give an example of a convergent sequence s_n with s_n > a for all n, but where lim_n→∞ s_n = a.