I've tried googling this one a bit but nothing seems to come up, even though its considered to be a well known fact.
Why does the kaprekar process of taking a 4 digit number: L, generating L' and L'' such that L' is the digits of L in ascending order and L'' is the digits in descending order and subtracting L' - L'' always converge to the Kaprekar constant of 6174?
Clearly the only value for which this process is constant is 6174 but that doesn't explain why there should be convergence.
One attempt at proof is to determine all the possible numbers that converge to 6174 after a single iteration, and then attempt to reason that each too can be reached by the convergence of even more numbers in such a way that if I continue this reasoning I should cover ALL 4 digit numbers.
But to turn my strategy into induction from brute force casework has become futile. What is the actual proof?