Why is the argument not applicable in proving rational numbers are uncountable?
The way I see it is-
Every rational number has a repeating decimal expansion. Now, let us say the number described in Cantor's argument repeats after 'n' digits. Any number in the list that doesn't have one of the first 'n' digits common with this number will not be the same as this number. But now, the Cantor argument breaks down for the (n+1)th place as this number cannot be different from any other rational number with the first n digits being common. So, as long as this number is not present in the list, the Cantor number will not be in the list. But if we map one of the natural numbers to a number with the first n digits common, then this cantor number will be in the list too, by nature of it being rational. Hence, we cannot use this to prove the rational numbers are uncountable. As it turns out, they are countable anyway.
Is this correct?
