Theorem 4.8: Every nonzero natural number n has a largest element.

Proof: We proceed by induction on n. If n = 0 the theorem is vacuously true. Assume that the theorem is true if n = k and we seek to show that it is true for n = k + 1.

To see that k is the largest element of k + 1, let

Either

in which case

by Theorem 4.7, whence x < k, by Definition 4.6, or

in which case

In either case, x < k.