For all real numbers, if x - floor(x) < 1/2, then floor(2x) = 2 * floor(x)

Proof:

Suppose that x is a real number and that x - floor(x) < 1/2

Since floor(x) is, by definition, always an integer, then x - floor(x) is the fractional part of x.

Note that if

x - floor(x) < 1/2

That is, fractional part of x is less than 1/2, then if we multiply the fractional part with 2 we won't get a fractional part greater than 1.

Hence it holds that:

floor(2x) = 2 * floor(x)

because the fractional part is always going to be less than 1.

Q.E.D.