Why is on the following link a limit superior of a sequence $$a_n=(-1)^n/n$$ defined separately for $n$ odd and even? Namely for $n$ odd its $A_n=1/(n+1)$ and for even its $A_n=1/n$. What does it even mean in this case when two sequences of supremums are defined?
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There is not two sequences of supremum defined. It is just that the value of $A_n = \sup_{k \ge n} a_k$ depends on $n$ odd or even. Then $\lim A_n =0$ and therefore $\limsup a_n =0$.
mathcounterexamples.net
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I don't understand, what is the point? $A_n = 0$ no matter if $n$ is even or odd. – Michael Munta Apr 25 '20 at 06:37
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My last comment was wrong. However the set of numbers considered contains strictly positive numbers. So its $\sup$ is strictly positive. – mathcounterexamples.net Apr 25 '20 at 07:22
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Not sure I understand what you mean.Maybe I don't understand the notation. What does this really mean? $A_n = \sup_{k \ge n} a_k$ – Michael Munta Apr 25 '20 at 07:53
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It means $\sup {a_n, a_{n+1}, \dots}$. – mathcounterexamples.net Apr 25 '20 at 08:55
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Ah, sorry. So $A_n$ is a single sequence that is defined differently for $n$ odd or even, I was thinking it meant something else. – Michael Munta Apr 25 '20 at 09:21