Edited in light of Gerry Myerson's quick counterexample.
I have two finite sequences $(a_n)$ and $(b_n)$ satisfying the following:
- all terms are positive
- both sequences are strictly decreasing
- the $(b_n)$ strictly dominates $(a_n)$ (i.e. $b_n > a_n$ for all $n$)
- the sequence $\left(\frac{a_n}{b_n}\right)$ is strictly decreasing
I want to conclude $ \sum_{n=1}^N (-1)^{n-1}a_n \leq \sum_{n=1}^N (-1)^{n-1}b_n. $
What additional properties might I seek to establish for the sequences to get the desired result?