Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 3 / d / iii / Solution

False. It suffices to find a subsequence along which the ratio is unbounded. Set $n=4m$. The last $3m$ factors of $(4m)!$ are at least $m$, while the first $m$ factors have product at least one. HenceThis contradicts the eventual uniform bound required by Big O notation. The proof is an elementary factorial comparison, independent of Stirling's approximation.