In the following version of Cantor's diagonal argument, where is the assumption that the nth digit of r must be different from 0 or 9 used? Thanks
Suppose f is a 1-1 mapping between the positive integers and the reals.
Let dn be the function that returns the n-th digit of a real number.
Now, let's construct a real number, r. For the n-th digit of r, select something different from dn(f(n)), and not 0 or 9.
Now, suppose f(m) = r. Then, the m-th digit of r must be dm(r) = dm(f(m)), but by construction that cannot be the m-th digit of r.
Therefore, no such f exists.
Thus, there is no 1-1 mapping between the positive integers and the reals.
The important thing to note is when I construct r, it really is a real number.
(by n-th digit, I mean to the right of the decimal point)