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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.

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

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