Consider the set of real "fractions", F: Let b (/b/>1), a_i (i = 1,n) be a finite set of reals and F = {x| x = Sum (j = -1, -2, ...,) of a_j X b^j }. Take as a hypothesis that every real may be represented uniquely by such a sum if we include "whole" numbers (terms with non-negative exponents of b) in the above sum.

I wish to show that F is a closed set. My efforts to date are to consider that y is an accumulation point (ap) of F . Thus, any interval that includes y, no matter how small, includes a point z of F, by definition of an ap. Thus, the distance between y and z may be smaller than any chosen positive epsilon. I try to take an esimate of that distance in order to show that y is a member of F and, therefore, F is closed. By hypothesis, y has a unique positional representation, but it must include the "whole" numbers as well as the "fractions". My estimate, does not seem to lend itself simply to dropping the terms with non-negative exponents of b (the radix). After all b may be a positive or negative real, though greater than 1 in absolute value.

If what I have written is clear enough, perhaps I could get a hint or two.