Kaprekar's Process on 5 Digit Number

Theorem
Let $n$ be a $5$-digit integer whose digits are not all the same.

Kaprekar's process, when applied to $n$, results in one of the following $3$ cycles:


 * $53 \, 955 \to 59 \, 994 \to 53 \, 955$


 * $61 \, 974 \to 82 \, 962 \to 75 \, 933 \to 63 \, 954 \to 61 \, 974$


 * $62 \, 964 \to 71 \, 973 \to 83 \, 952 \to 74 \, 943 \to 62 \, 964$

Proof
We have: