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Real Analysis : Limit Superior

Let a_n be bounded sequence.prove that a-the sequence defined by y_n=sup{a_k:k>=n} converges.

b- Prove that lim inf a_n<=lim sup a_n for every bounded sequence and give example of a sequence which the inequality is strict.

Solution Preview

a. Since a_n is bounded, say |a_n|<=M for all n and some M>0. y_n=sup{a_k:k>=n}, then |y_n|<=M. So y_n is bounded. We note the set {a_k:k>=n+1) is contained in the ...

Solution Summary

Limit Superior is investigated.