It is well-known that $(-1)^j \sum_{i=0}^j (-1)^i\binom{n}{i} \geq 0$. This inequality can be used to prove Bonferroni's inequalities for example. Recently I noticed that a similar inequality applies to non-negative unimodal sequences with alternating zero sum: if $a_0 \leq a_1 \leq \ldots \leq a_k \geq a_{k+1} \geq \dots \geq a_{n}$ with $\sum_{i=0}^n (-1)^i a_i = 0$ then $(-1)^j \sum_{i=0}^j (-1)^i a_i \geq 0$ for $j=0 \ldots n$. Of course the binomial sequence is just a special case. My question is if anyone has any knowledge of this property of unimodal sequences having appeared or having been used anywhere in the literature? Thanks.

Jose A Rodriguez