Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 214 / 1 / a / Solution

PutFix $k\ge 1$ and write $n=qk+r$ with $0\le r<k$. Repeated subadditivity givessoThe finitely many values $x_0,\dots ,x_{k-1}$ are bounded, henceTaking the infimum over $k$ gives $\mathrm{limsup}{x}_n/n\le \alpha$, while the definition of $\alpha$ gives $x_n/n\ge \alpha$ for every $n$. ThereforeThis is Fekete lemma.