Argument: Counting Pennies
Lemma 4: For any FIND(x) operation, the total number of pennies deposited (American and Canadian) is equal to the number of nodes on the path from x to its root r.
Lemma 5: Since we are performing a sequence of m FIND operations, the number of American pennies deposited is at most m(G(n) + 2),
because in any FIND operation, 2 American pennies are deposited for the root and its child, and from x to r the ranks increase monotonically and there are only G(n) rank groups offering only G(n) opportunities to deposit an American penny according to the rule.