For any non-negative integer observe that
Thus, if we have
There are infinitely many non-negative integers and hence infinitely many such rationals
Note that for any Hence, in order to have we must have If is a positive non-integer rational, would have more decimal places than This would mean that their fractional parts cannot add up to 1. Hence there are no positive rationals such that
Remark: There do however exist positive irrational numbers with To see this, note that is continuous on the interval and while the result follows by the intermediate-value theorem.