Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 204 / 1 / Solution

Let $\alpha ={\inf }_{r\ge 1}{a}_r/r$. The inequality $a_n/n\ge \alpha$ gives ${\mathrm{liminf}}_n{a}_n/n\ge \alpha$. Fix $r$ and write $n=qr+s$, where $0\le s<r$. Repeated subadditivity gives $a_n\le q{a}_r+a_s$ when $s>0$, with the evident omission when $s=0$. Since the finitely many remainders $a_s$ are bounded,Taking the infimum over $r$ proves the Fekete lemma: