I was trying to make a problem, but I realized that the hypotheses are wrong, as for example, $ a_n =-n! $ Satisfies the hypothesis, but the alternating sum is clearly bounded. Suppose that the hypothesis was that if it were limited, in this case, as it could prove that the remainder of the lim sup and lim inf, given that value?
I suppose I must calculate separately each cleaning, the problem is I do not know very well as you would
I realize that the sum of the pair is growing, and the odd is decreasing, but among them are not if they can be compared. But for example if $ a_2 $ is positive then have the following $ ...
The problem is : Let $ a_n $ a non increasing sequence , let $ s_n = \sum\limits_{k = 1}^n {a_k \left( { - 1} \right)^{k + 1} } $ prove that $ s_n $ it´s bounded ( false, by the counterexample, but assume that it´s bounded) and also prove that $ \lim \sup s_n - \lim \inf s_n = \lim a_n $
Reading the end I realized that $ a_n $ is bounded (because the last thing they ask me to prove assume that a_n limit exists)