Kaprekar's Process for 2-Digit Numbers

Theorem
Kaprekar's process, when applied to a non-repdigit $2$-digit positive integer leads to the cycle:
 * $09 \to 81 \to 63 \to 27 \to 45 \to 09$

Note that it is important to retain the leading zero on the $9$, or the process trivially terminates in $0$.

Proof
Let $n \in \Z_{>0}$ be a $2$-digit positive integer.

, let $n$ be expressed in decimal notation as:
 * $n = 10 x + y$

where:
 * $x > y$
 * $0 \le x \le 9, 0 \le y \le 9$

The reversal of $n$ is $10 y - x$.

We have:

Thus after the first iteration of Kaprekar's process, $n$ is one of:
 * $09$
 * $18$
 * $27$
 * $36$
 * $45$

or their reversals.

Applying Kaprekar's process to $09$ gives:

All those multiples of $9$ can be seen to be in (or end up in) this loop.

Hence the result.