Exercise 1.6.3

Supply rebuttals to the following complaints about the proof of Theorem 1.6.1.

(a)

Every rational number has a decimal expansion, so we could apply this same argument to show that the set of rational numbers between 0 and 1 is uncountable. However, because we know that any subset of $Q$ must be countable, the proof of Theorem 1.6.1 must be flawed.

(b)

Some 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 $.4999\dots$ Doesn’t this cause some problems?