TR-MO 1995 Problem 3

Let $A$ be a real number and $(a_{n})$ be a sequence of real numbers such that $a_{1}=1$ and $$1<\frac{a_{n+1}}{a_{n}}\leq A \text{ for all }n\in\mathbb{N}.$$ $(a)$ Show that there is a unique non-decreasing surjective function $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $1<A^{k(n)}/a_{n}\leq A$ for all $n\in \mathbb{N}$ . $(b)$ If $k$ takes every value at most $m$ times, show that there is a real number $C>1$ such that $Aa_{n}\geq C^{n}$ for all $n\in \mathbb{N}$ .