Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 3 / e / iii / Solution

Take $f(n)=g(n)=n^2$, so $h(n)=n^2$ but $(f\circ g)(n)=n^4$. The ratio $(f\circ g)(n)/h(n)=n^2$ is unbounded, precluding any eventual constant upper bound in big O notation. False: in general $f\circ g\ne O(h)$. This is a counterexample involving function composition, not pointwise addition.