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:$$\begin{aligned}O(g(n)) = \{f(n):&\text{ there exist positive constants $c$ and $n_0$ such that}\\&0 \le f(n) \le cg(n) \text{ for all } n \ge n_0\}\end{aligned}$$
Does the transitivity of big-O notation hold for asymptotically nonnegative functions?
asymptotics;landau notation