Prove that for all positive integers n, p(n) if and only if p(1) (all positive integers n, p(n) p(n + 1)) for any given predicate p. (Hints: If p(1) (all positive integers n, p(n) p(n + 1)), then for all positive integers n, p(n). For the only if part, you can assume p(1) (all positive integers n, p(n) p(n + 1)) is false and prove for all positive integers n, p(n) is false, too. It's the contrapositive of the only-if part.)