Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 3 / b / ii / Solution

False. Set $r_n=n!/2^n$. The ratio of successive terms isConsequently $r_n\ge r_3{2}^{n-3}$ for $n\ge 3$, which is unbounded. No constant can bound $n!$ by that constant times $2^n$ for all sufficiently large $n$. The factorial therefore fails the proposed Big O notation bound, without needing Stirling's formula.