A Craps Tutorial

Other Topics Section    --    Bracket Function Properties

	Theorem   5
		If x is any real number then - -x   is the smallest integer greater than or equal to x

Theorem 5 is obviously true if x is an integer.
So, let's assume that x is not an integer.

Then there must exist a real number   α   such that
x   =   x   +   α      and   0   <   α   <   1.

Then   -x   =   - x   +   ( -α )      where   -1   <   -α   <   0.

Now use the trick of adding and subtracting the same value to some item.
Here we subtract and add 1 to the right hand side of the above equation.

-x   =   -1 - x   +   ( 1 - α )    where   0 < 1 - α   <   1

	By theorem 1,	-x	=	-1   -   x   +   1 - α	statement 1
			=	-1   -   x	statement 2
	Therefore,	- -x	=	1   +   x	statement 3
	x not an integer	→	x must lie between two consecutive integers
		→	x   <   x   <   1   +   x
	Since   1 + x   is obviously the smallest integer greater than x,
	and since,   by statement 3,   1 + x   =   - -x , we are done with this case.