Let $ a$ satisfy the equation \[ \frac {1}{a} + \frac {1}{a + k} = 1, \] i.e., \[ a = \frac {\sqrt {k^2 + 4} - k + 2}{2}. \] Since $ k$ is a positive integer, it is clear that both $ a$ and $ a + k$ are irrational and thus produce a pair of complementary Beatty sequences $ y_n = \lfloor an\rfloor$ and $ \lfloor (a + k)n\rfloor = y_n + kn$. These sequences form a partition of the positive integers, implying that each $ y_n$ is different from any of $ y_m + km$. But this matches the definition of the sequence $ x_n$, implying that $ x_n = y_n = \lfloor an\rfloor$ (a rigorous proof can be done by induction on $ n$).