Appendix A.2
Theorem: Every ball on a pool table is the same color as every other ball on the table.
Proof: Let n be the number of balls on the table. The statement is true if n = 1, because the one ball is the same color as itself. Now assume n > 1 and the statement is true for 1, 2, 3,..., n - 1. Pick one of the balls and remove it from the table. Now there are n - 1 balls on the table, so by the assumption they are all the same color. Put the ball you removed back, and pick up a different ball. Again, all the balls on the table are the same color by the inductive assumption. So both of the balls you removed are the same color as all the other balls on the table; hence the statement is true for n.