Why does the Θ-class survive adding a constant only for positive, monotonic, and non-decreasing functions? I know that for positive monotonically non-decreasing functions, f(n) and g(n), 

f(n) = O(g(n) + c) entails
f (n) = O(g(n))


Why is this always true only for positive monotonically non-decreasing functions? $\Theta$

If there exists one, give a counter-example that shows that the above Big O rule is not necessarily true for functions that are not monotonically non-decreasing.

I'm really confused why the rule specifies only positive, monotonic, non-decreasing functions. Thanks for your help!
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.