SIMON-MARAIS 2021 A4

For each positive real number $r$ , define $a_0(r) = 1$ and $a_{n+1}(r) = \lfloor ra_n(r) \rfloor$ for all integers $n \ge 0$ . (a) Prove that for each positive real number $r$ , the limit $$L(r) = \lim_{n \to \infty} \frac{a_n(r)}{r^n}$$ exists. (b) Determine all possible values of $L(r)$ as $r$ varies over the set of positive real numbers. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ .