Chapter 2.5, Problem 32E

In constructing the number x in Example 4, how would you decide what to put in the 99th place?

Example 4A Cardinal Number Greater Than ${\aleph }_0$

We will reproduce Cantor’s argument that the seat of real numbers between 0 and 1 has cardinal number greater than ${\aleph }_0$. We begin by assuming that we can put the set of numbers between 0 and 1 in a one-to-one correspondence with the natural numbers and show that no matter how hard we try, there will always be some number that we could not leave listed.

Although we would not actually know what the listing would be, for the sake of argument, let us assume that we had listed all the numbers between 0 and 1 as follows:

$\begin{array}{l}1\leftrightarrow 0.6348291347\cdots \\ 2\leftrightarrow 0.2373261008\cdots \\ 3\leftrightarrow 0.4821063391\cdots \\ 4\leftrightarrow 0.6824537128\cdots \\ 5\leftrightarrow 0.4657189233\cdots \\ ⋮⋮\end{array}$

Although we have assumed that all numbers between 0 and 1 are listed, we will now show you how to construct a number x between 0 and 1 that is not on this list. We want x to be different from the first number on the list, so we will begin the decimal expansion of x with a digit other than a 6 in the tenths place, say x = 0.5…Because we don’t want x to equal the second number on the list, we make the hundredths place not equal to 3, say 4, so far, x=.54…(For this argument to work, we will never switch a number to 0 to 9.) Continuing this pattern, we make sure that x is different from the third number in the third decimal place, say 3 and different from the fourth number in the fourth decimal place, make it 5, and so on. At this point, x = 0.5435… By constructing x in this fashion, it cannot be the first number on the list, or the second, or the third, and so on. In fact, x will differ from every number on the list in at least one decimal place, so it cannot be any of the numbers on the list. This means that our assumption that we were able to match the numbers between 0 and 1 with the natural number is wrong So the cardinal number of this set is not ${\aleph }_0$.