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:

$\ds 95 \, 553 - 35 \, 559$

$=$

$\ds 59 \, 994$

$\ds 99 \, 954 - 45 \, 999$

$=$

$\ds 53 \, 995$

$\ds 97 \, 641 - 14 \, 679$

$=$

$\ds 82 \, 962$

$\ds 98 \, 622 - 22 \, 689$

$=$

$\ds 75 \, 933$

$\ds 97 \, 533 - 33 \, 579$

$=$

$\ds 63 \, 954$

$\ds 96 \, 543 - 34 \, 569$

$=$

$\ds 61 \, 974$

$\ds 96 \, 642 - 24 \, 669$

$=$

$\ds 71 \, 973$

$\ds 97 \, 731 - 13 \, 779$

$=$

$\ds 83 \, 952$

$\ds 98 \, 532 - 23 \, 589$

$=$

$\ds 74 \, 943$

$\ds 97 \, 443 - 34 \, 479$

$=$

$\ds 62 \, 964$

This theorem requires a proof.
In particular: It remains to be shown that all $5$-digit numbers end up here
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Sources

1972: Boris A. Kordemsky: The Moscow Puzzles: 359 Mathematical Recreations: $\text {XIII}$: Numbers Curious and Serious: $350$. A Persistent Difference
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $99,954$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $99,954$