Separate tails of sequences in defining lim sup/inf

Question

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?

Answer 1

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 $\limsup a_n =0$.