ÖMO 2006 Problem 4

Let $<h_n>$ $n\in\mathbb N$ a harmonic sequence of positive real numbers (that means that every $h_n$ is the harmonic mean of its two neighbours $h_{n-1}$ and $h_{n+1}$ : $h_n=\frac{2h_{n-1}h_{n+1}}{h_{n-1}+h_{n+1}}$ ) Show that: if the sequence includes a member $h_j$ , which is the square of a rational number, it includes infinitely many members $h_k$ , which are squares of rational numbers.