<s>[INST] Let \(f:\mathbb{Z} \rightarrow\mathbb{R}\) be a function that satisfies the following properties:
\(-\)   For every integer \(x\), \(f(x) + f(x+1) > 0\).
\(-\)   For every pair of integers \(x\) and \(y\) with \(x < y\), \(f(x) + f(x+1) + \dots + f(y) \ge 0\).
\(-\)   There exists an integer \(x\) such that \(f(x) < 0\).

Prove that there exist infinitely many integers \(x\) such that \(f(x) > 0\).[/INST]