Prove the principle of strong induction: Let P (n) be a statement that is either true or false for each n ∈ N. Then P (n) is true for all n ∈  provided that (a) P (1) is true, and (b) for each k ∈ N,...

Prove the principle of strong induction: Let P (n) be a statement that is either true or false for each n ∈ N. Then P (n) is true for all n ∈
 provided that

(a) P (1) is true, and

(b) for each k ∈ N, if P ( j) is true for all integers j such that 1 ≤ j ≤ k, then P (k + 1) is true.