I'm reading The Foundations of Mathematics and trying to understand the proof of the finite and infinite representation of the same real number.
For example, if a number x has two different decimal expansions, then, without loss in generality, we can take $x = a_0 · a_1 . . . a_{n–1}a_n . . . = a_0 · a_1 . . . a_{n-1}b_n . . .$ where $a_n < b_n$.
Multiply through by $10^n$ to get $a_0a_1 . . . a_{n–1}a_n · a_{n+1} . . . = a_0a_1 . . . a_{n–1}b_n · a_{n+1}. . .$ where $a_n < b_n$.
Subtracting the whole number $a_0a_1 . . . a_{n–1}a_n$ gives $0 · a_{n+1} . . . = k · b_{n+1}$ . . . where $k = b_{n+1} – a_{n+1} > 0$ is a positive integer.
But the first decimal is $0 · a_{n+1} . . . < 0 · 999 . . . ≤ 1$ and the second exceeds the positive integer k. So they can be equal only if $k = 1$ and both decimals represent the same limiting value 1. In this case, $a_{n+1} = a_{n+2} = · · · = 9, b_{n+1} = b_{n+2} = · · · = 0$ and $b_n = a_n + 1$.
Specifically, how do they produce $0 · a_{n+1} . . . = k · b_{n+1}$ . . . where $k = b_{n+1} – a_{n+1} > 0$ is a positive integer? Shouldn't it be $0 · a_{n+1} . . . = k · a_{n+1} . . .$ where $k = b_{n} – a_{n}$ ?
Also which decimals are meant by "the first" and "the second"?