Real Analysis Suppose {an} is a bounded sequence of Real numbers. If Lim(Supan)=M as n approaches infinity prove that for every Epsilon>0, we have an<M+Epsilon for all but a finite number of values of n. I am told that this is the ratio test using Limit Supremum. I don't understand this statement. If M is the Lim of the Supremum, then how can some element of an be between M and M+Epsilon? Isn't the Supremum the Greatest Lower Bound? Meaning that there are no elements above the Supremum?
Real Analysis Suppose {an} is a bounded sequence of Real numbers. If Lim(Supan)=M as n approaches infinity prove that for every Epsilon>0, we have an<M+Epsilon for all but a finite number of values of n. I am told that this is the ratio test using Limit Supremum. I don't understand this statement. If M is the Lim of the Supremum, then how can some element of an be between M and M+Epsilon? Isn't the Supremum the Greatest Lower Bound? Meaning that there are no elements above the Supremum?
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 77E
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Suppose {an} is a bounded sequence of Real numbers. If Lim(Supan)=M as n approaches infinity prove that for every Epsilon>0, we have an<M+Epsilon for all but a finite number of values of n.
I am told that this is the ratio test using Limit Supremum. I don't understand this statement. If M is the Lim of the Supremum, then how can some element of an be between M and M+Epsilon? Isn't the Supremum the Greatest Lower Bound? Meaning that there are no elements above the Supremum?
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