TITLE: How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals?
QUESTION [5 upvotes]: How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals?

REPLY [7 votes]: This follows immediately from the Archimedian property. To find a dyadic $\rm\ m/2^n \in (r,s)\ $ start at at any dyadic $\rm\ k < r\ $ (e.g. an integer) and keep taking dyadic step sizes smaller than the interval, say $\rm\ 1/2^j\ <\ s-r\:.\: $ By the Archimedean property eventually you'll land in the interval, necessarily at some dyadic rational (being a sum of such). This is a special case of the proof I explained here.