Suppose you take an open interval I of length 1, divide it into countable sub-intervals (I/2, I/4, etc.), and cover each rational with one of the sub-intervals.
Since all the rationals are covered, then it seems that sub-intervals (if they don't overlap) are separated by at most a single irrational. For instance, if s = sqrt(2) then we might have the two sub-intervals ( s-I/2, s ) and ( s, s+I/4 ) and s would be uncovered.
But if we have a countable number of open sub-intervals with at most one (irrational) number between each sub-interval, that would mean we've covered all of the real line except for the complement, a countable number of irrationals, with an interval of length 1, which is absurd.
So there's a flaw in the reasoning but I don't see exactly where. Where is it?