Why does the Θ-class survive adding a constant only for positive, monotonic, and non-decreasing functions?
Let $f(n) = 1-1/n$ and $g(n) = 1/n$, so we've violated the condition that $g$ is non-decreasing. However, it's not hard to show that $f(n) \in O(g(n)+2)$ but we also have $f(n)\notin O(g(n)$, invalidating the inference.