The title says it all; I'm wondering if there is a sequence $\{f(n)\}_{n<\omega}$ of negative integers such that successive terms get strictly larger.
The issue seems to be that as soon as we fix $f(0)=-n$ we can only strictly increase $n$ times due to the discreteness of $\mathbb{Z}^-$, however my intuition is that the cardinality of $\mathbb{Z}^-$ being $\omega$ will admit such a sequence non-constructively by choice.