The Real and Ordinal Number

For n > 0, we double z = {x - (1/2") | x + (1/2")} to see that 2z is the simplest number between z + x - (1/2") and z + x + (1/2"). Since z certainly lies between x — (1/2") and x + (1/2") these limits are between 2x — (1/2"_1) and 2x + (\/2 '~ l\ and by induction 2x is the simplest number between these limits, so that 2z = 2 x ,z = x.