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 / iv / Solution

For each fixed positive integer $k$, part (i) gives $(f(n)+g(n){)}^k\le (2h(n){)}^k=2^{k}h(n{)}^k$. This meets big O notation with constant $2^k$ and threshold $1$. True: $(f+g{)}^k=O(h^k)$ for fixed $k$. The constant may depend on $k$; no claim uniform in a varying $k$ is required.