Chapter 3.3, Problem 21MS

Nines. Would Cantor’s argument work if we used 2 and 9 instead of 2 and 4 as the digits? That is, for each digit we ask whether the digit is a 2. If it is 2, we make the analogous digit of M a 9, and otherwise we make the digit a 2. Using this method, we are not guaranteed that the number M we construct is not on our list. Provide a scenario in which the constructed number is on the list! (Hint: Remember from Chapter 2 that $\mathrm{0.1999999...}=\mathrm{0.2.}$ The number 9 is key here.)