Given that:
$p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+⋯$
Why can one not state:
$p(n)≥p(n-1)+p(n-2)-p(n-5)-p(n-7)$
Here is the logic: the subsequent 2 terms of the relation are additive and the 2 following one are negative, etc. yet the magnitude of p(k) steadily decreases hence the resultant value most be slightly positive for all of the following double pairs and for an $n$ large enough, it is not even necessary to consider the last few terms of the sum in terms of their potential to tip the inequality.
Where is the mistake?