Let $$S$$ be the set of all real numbers $$r < 1000$$ whose hexadecimal (base 16) expansions do not contain any letters (i.e. they only have digits in the set $$\{0, 1, 2, ... 9\}$$).

Find the least upper bound of $$S$$.

Details:

For example, $$1.45 \in S$$ because its hexadecimal expansion is $$1.7\overline{3}_{16}$$, which only consists of the digits $$1,3,7$$. However $$240 \not \in S$$ because its hexadecimal expansion is $$f0_{16}$$ which contains an $$"f"$$.