Does the transitivity of big-O notation hold for asymptotically nonnegative functions?

Question

I'm reading the book Introduction to Algorithms and in Chapter 3 it is said that if $f(n)$ and $g(n)$ are asymptotically positive then

$$ f(n) = O(g(n)) \text{ and } g(n) = O(h(n)) \text{ implies } f(n) = O(h(n)) $$

However, I don't see why this wouldn't also be valid if you just required $f(n)$ and $g(n)$ to be asymptotically nonnegative.

Edit: Here is the definition of $O$ used in the book:

$$ \begin{aligned} O(g(n)) = \{f(n):&\text{ there exist positive constants $c$ and $n_0$ such that}\\ &0 \le f(n) \le cg(n) \text{ for all } n \ge n_0 \} \end{aligned} $$

Answer 1

Suppose $f(n)$, $g(n)$ and $h(n)$ are asymptotically non-negative functions such that $f(n) = O(g(n))$ and $g(n) = O(h(n))$.

Then, by definition, there exist positive constants $c_1, c_2, m_1, m_2$ such that

$$ 0 \le f(n) \le c_1 g(n) $$

whenever $n \ge m_1$ and

$$ 0 \le g(n) \le c_2 h(n) $$

whenever $n \ge m_2$.

Choose $c = c_1 c_2$ and $n_0 = \max\{m_1, m_2\}$. Note that both $c$ and $n_0$ are positive. Then, whenever $n \ge n_0$, we have

$$ 0 \le f(n) \le c_1 g(n) \le c_1 c_2 h(n) = c h(n) $$

which implies that $f(n) = O(h(n))$.