Question 1.

Part (1) Prove that the difference of two odd integers is even. Give a justification at each step.

Part (II) Prove that the sum of any two rational numbers is a rational number. Give a justification at each step.

Part (III) Prove by contradiction that there is no greatest integer.

Part (IV) Prove by contraposition that for all integers n, if n² is even then n is even.

Part (V) Prove using mathematical induction: integers n 1, 2 + 4 + ... + 2n = n² + n. Give justifications for each step.