Proofs by Case (3)

(3) If n is an even integer, then n = 4j or n = 4j - 2 for some integer j.

Proof. Assume that n is an even integer. Then n = 2k for some integer k.
Case 1: Assume that k is even. Then k = 2p for some integer p. So in this case

n = 2k = 2(2p) = 4p

and the results holds.

Case 2: Assume that k is odd. Then k = 2p + 1 for some integer p. So in this case

n = 2k = 2(2p + 1) = 4p + 2 + 4 - 4 = 4(p - 1) - 2 = 4j - 2

where j = p - 1 is an integer, and therefore the result holds.

So in all cases n = 4j or n = 4j - 2 for some integer j.

QED