Argument: Counting Canadian Pennies
Whenever a Canadian coin is deposited into a node, it will get a new parent of higher rank by path compression.
A node in rank group g > 0 can be moved no more than F(g) - F(g - 1) times, where F(g) = greatest rank in rank group g.
After this many moves, all further payments for the node use American pennies. (Its parent must then be in a different rank group.)
Lemma 6: The number of nodes N(g) in rank group g > 0 is no more than n/2F(g-1) .
To verify this, sum up the maximum number of nodes possible in each rank for ranks in the group.