Chapter 3.3, Problem 17MS

Pairs (S). In Cantor’s argument, is it possible to consider pairs of digits rather than single digits? That is, suppose we look at the first two digits of the first real number on our list; and, if they are not 22, then we make the first two digits of M be 22. If the first two digits are 22, then we make the first two digits of M be 44. Similarly, let the next two digits of the next real number on our list determine the next two digits of M, and so on. If this procedure would still produce a number M not on our list, then provide the details for such a method. If this procedure does not work, explain or illustrate why it does not.