I need help writing out the following proofs. Thank you.

1. Prove that there exists a natural number that satisfies 3 + 4 = 10.

2. Prove that there exist natural numbers and such that = 13.

3. Prove that for all real numbers , the inequality 2 + 2 2 holds.

4. Prove that there does not exist a natural that satisfies 3 + 4 = 5. [Your proof must only use integers, inequalities and elementary logic. You may use that adding a number to an inequality, or multiplying an inequality by a positive number, preserves the inequality. There must be no reference to rational numbers or quotients.]

5. Prove that for every integer , there exists an integer such at 2 + = 10.

6. Disprove that there is an integer so that for every integer , 2 + = 10.

7. Prove that there is an integer such that for every positive integer , ( 3) .

8. Prove that for every even integer , there exists an odd integer with < + 5 < + 3.