Sequence of Limit Superiors Bounded and Decreasing

I am aware that the limit superior is equivalent to the least upper bound of sequences, but why is it that the sequence of limit superiors is bounded and decreasing? I can't intuitively understand this -- I understand that as there are more and more terms in the sequence, then the limit superior will tend to the least upper bound of the whole sequence, but I think it should be an increasing sequence, not decreasing.

Would it be possible to provide an example?

Thanks.

• To understand lim sup, think for example of the sequence $a_n=(-1)^n\left(1+\dfrac1n\right)$ – J. W. Tanner Sep 4 '19 at 23:25

1 Answer

Your question is badly worded but I will try to guess what you are saying. Let $$(a_n)$$ be a bounded sequence of real numbers and let $$b_n=\sup (\{a_k: k \geq n\})$$. The sequence $$(b_n)$$ is decreasing: this is because $$(\{a_k: k \geq n+1\}) \subseteq (\{a_k: k \geq n\})$$ which implies $$b_{n+1} \leq b_n$$: if $$A \subseteq B$$ then $$\sup A \leq \sup B$$.