1.35 Remark [CANTOR'S DIAGONALIZATION ARGUMENT]. The open interval (0, 1)

is uncountable.

STRATEGY. Suppose to the contrary that (0,1) is countable. Then by definition, there is a function f on N such that f(I), f(2), ... exhausts the elements of (0,1).

We could reach a contradiction if we could find a new number x E (0,1) that is different from all the f(k)'s. How can we determine whether two numbers are different? One easy way is to look at their decimal expansions. For example, 0.1234 =f. 0.1254 because they have different decimal expansions. Thus, we could find an x that has no preimage under f by making the decimal expansion of x different by at least one digit from the decimal expansion of EVERY f(k).

There is a flaw in this approach that we must fix. Decimal expansions are unique except for finite decimals, which always have an alternative expansion that terminates in 9's, e.g., 0.5 = 0.4999 ... and 0.24 = 0.23999 ... (see Exercise 10, p. 44).

Hence, when specifying the decimal expansion of x we must avoid decimals that terminate in 9's.