Question 97
Let $\{a_{n}\}_{n=1}^{\infty}$ be a sequence of integers greater than $1$ and let $x\gt 0$ be an irrational number. Denote by $x_{n}$ the fractional part of the product $a_{n} a_{n-1} \ldots a_{1} x$
a) Prove that $x_{n} \gt \frac{1}{a_{n+1}}$ for infinitely many $n$.
b) Find all sequences $\{a_{n}\}_{n=1}^{\infty}$ such that there exist infinitely many $x \in (0,1)$ for which $x_{n} \gt \frac{1}{a_{n+1}}$ for all $n$.
a) Prove that $x_{n} \gt \frac{1}{a_{n+1}}$ for infinitely many $n$.
b) Find all sequences $\{a_{n}\}_{n=1}^{\infty}$ such that there exist infinitely many $x \in (0,1)$ for which $x_{n} \gt \frac{1}{a_{n+1}}$ for all $n$.