341Suppose there are m ≤ n committees where n is the number of senators
Let the sum of the absolute ratings of senators in committee i be represented by si
Suppose member k with abs rating xk is permitted to migrate. He will migrate to another committe sj if we have si-sj>xk (this follows from the condition for moving given)
Now consider the function F = Σ (sa-sb)2 summed over all pairs of committees. (you can also consider Σ |sa-sb|
The claim is that whenever a migration takes place this function decreases.
To see how, first consider the two committees between which the migration takes place. Prior to migration the summand is (si-sj)2 and subsequent to migration it is (si-sj-2xk)2 and since we have si-sj>xk it follows that (si-sj)2 > (si-sj-2xk)2
Again consider any other committee st
Prior to migration the summand is (si-st)2 +(sj-st)2 and subsequent to migration it is (si-st-xk)2 +(sj-st+xk)2 and again since si-sj>xk we have (si-st)2 +(sj-st)2 > (si-st-xk)2 +(sj-st+xk)2
Thus we have a proved that the function F decreases with migration. But F takes only integer values and is bounded below by zero. Hence this migration cannot take place infinitely and will stop.