Supply rebuttals to the following complaints about the proof of theorem 1.5.1.
(a) every rational numbers has a decimal expansion so we could apply this same argument to show that the set of rational numbers between 0 and 1 is un- countable. however, because we know that any subset of q must be countable, the proof of theorem 1.5.1 must be flawed.
(b) a few numbers have two different decimal representations. specifically, any decimal expansion that terminates can also be written with repeating 9's. for instance, 1/2 can be written as .5 or as . doesn't this cause some problems?