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

2. Prove that the sum of any two rational numbers is a rational number. Give a justification at each step.

3. Prove by contradiction that there is no greatest integer.

4. Prove by contraposition that for all integers n, if n 2 is even then n is even.

5. Prove using mathematical induction: integers n 1, 2 + 4 + ... + 2n = n 2 + n. Give justifications for each step.