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 ° and 1 is uncountable. 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 .4999 Doesn't this cause some problems?