Why the two distribution are the same

Question

I am very confused with two distribution:

1. $\ln y = \ln S-rb$, where $b\sim$ Uniform(0,1)
2. $\ln y = r(⌊\frac{\ln S}{r}+\beta⌋ - \beta)$, where $\beta\sim $Uniform(0,1)

why $\ln y \sim $Uniform$(\ln S-r, \ln S)$?

For Equation 2, we have $\ln S-r \le \ln y\le \ln S$, but I cant understand the relationship between the two distribution.

Thanks!