BALTIC-WAY 2025 Problem 2

Let $a_1, a_2, \ldots$ be a sequence of real numbers such that

$$\{a_n\} \lfloor a_{n+1} \rfloor = \lfloor a_n \rfloor \{a_{n+1}\}$$

for each positive integer $n$. Prove that there is a real number $\lambda$ such that $\{a_m\} \lfloor a_m \rfloor = \lambda$ for infinitely many positive integers $m$.

Remark: For a real number $x$, $\lfloor x \rfloor$ denotes the greatest integer that does not exceed $x$, and $\{x\} = x - \lfloor x \rfloor$.