Problem #PRU-111829

Problem

In the infinite sequence $(x_n)$, the first term $x_1$ is a rational number greater than 1, and $x_{n+1}=x_n+\frac{1}{\lfloor x_n\rfloor }$ for all positive integers $n$.

Prove that there is an integer in this sequence.

Note that in this problem, square brackets represent integers and curly brackets represent non-integer values or 0.