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?

Question

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?

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