AUSTRIAN-POLISH 1984 Problem 8

The functions $f_0,f_1 : (1,\infty) \to (1,\infty)$ are given by $f_0(x) = 2x$ and $f_1(x) =\frac{x}{x-1}$ . Show that for any real numbers $a, b$ with $1 \le a < b$ there exist a positive integer $k$ and indices $i_1,i_2,...,i_k \in \{0,1\}$ such that $a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b$ .