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

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

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?