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

Prove the explicit bound $n!\le 2^{n^2}$ by mathematical induction. At $n=1$, $1\le 2$. For the step, note that $n+1\le 2^n$ for $n\ge 1$: this auxiliary inequality has base $2=2$ and follows at the next index from $n+2\le 2(n+1)$. Hence, using the induction hypothesis,Thus $n!=O((2^n{)}^n)$, with constant $1$ and threshold $1$ in big O notation.