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Advanced MathQ&A LibraryReal 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?Start your trial now! First week only $4.99!*arrow_forward*

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Real Analysis

Suppose {a_{n}} is a bounded sequence of Real numbers. If Lim(Supa_{n})=M as n approaches infinity prove that for every Epsilon>0, we have a_{n}<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 a_{n} 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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