 # Let on be a bounded sequence. In this workshop, we will derive seve...

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Let on be a bounded sequence. In this workshop, we will derive several results about the numbers called lim inf an and lim supan- Let. us define new sequences bn and Cn as follows: In other words, bn is the greatest lower bound of the set {ax : kej and le n} and Gn is the least upper bound of the same set {at : * € J and le n). Problem (1) Prove that on is a bounded monotone sequence. This implies that bn converges to some real number. This real number is called lim inf an. In other words, ba converges to liminfan- Problem (2) Prove that Cn is a bounded monotone sequence. This implies that Cn converges to some real number. This real number is called limsupan- In other words, Cn converges to lim sup an- Problems (1) and (2) give us the definitions of lim inf an and lim supon- Now we can prove some theorems about these concepts: Problem (3) Prove the existence of a suhsequence of an such that this subsequence converges to lim inf an- Problem (4) Prove the existence of a subsequence of an such that this subsequence converges to lim sup on- Problem (5) Prove: If an has a subsequence that converges to a, then lim inf on <05 limsupún- Problem (6) Prove: If liminfan = limsupan then an converges to a, where a = lim inf an = lim supan- Problem (7) Prove: If an converges to some a, then lim inf an = lim sup On. The result in Problem (6) then implies a = liminfan = lim sup on-

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