Problem 2. [20 points, 10 points each]
a) Prove that $a^2 \ge a$ whenever $a$ is a natural number using proof by cases. Hint: You have two
cases: (1) $a = 0$ and (2), $a \ge 1$
b) Prove that if $a > 0$ and $b > 0$ then their geometric mean, $\sqrt{ab}$, is at least as large as $min(a, b)$.
Remember \"without loss of generality\". You can use the fact that $\sqrt{}$ is increasing (i.e. if $x \le y$
then $\sqrt{x} \le \sqrt{y}$).
Problem 3. [20 points, 10 points for each part] Recall the definition of absolute value:
$|x| = \begin{cases} x & \text{if } x \ge 0\\ -x & \text{otherwise.} \end{cases}$
a) Prove that $|x| = |-x|$ for all real numbers.
b) Prove that $|x - y| = |y - x|$.
Problem 4. [10 points] Prove or disprove the following statements:
a) [3 points] There exists a unique natural number $x$ such that $x^2 = x$.
b) [3 points] The product of two irrational numbers is always irrational.
Hint: You can use the fact $\sqrt{2}$ is irrational
c) [4 points] The sum of two irrational numbers is always irrational.
