Simple Proof Methods

Fun class activities:

1. Prove that if the integers $a$ and $b$ are each divisible by $d$, then $a + b$ is also divisible by $d$.
2. Prove that if $m$ is a multiple of $4$, then $m^2$ is a multiple of $16$.
3. Prove that if $m$ is a multiple of $4$, then $m$ is a multiple of $2$, too.
4. Prove that $n^4 - n^2$ is even for all $n \in \mathbb{N}$ (hint: use a proof by cases: (1) $n$ is even, and (2) $n$ is odd.)
5. Prove the triangle inequality: $|x + y| \le |x| + |y|$ for any $x, y \in \mathbb{R}$ (hint: use a proof by cases.)
6. Prove that $|xy| = |x|\cdot|y|$ for any $x, y \in \mathbb{R}$ (hint: use a proof by cases.)
7. Prove that $|x| = \sqrt{x^2}$ for any $x \in \mathbb{R}$ (hint: use a proof by cases.)

---

We have two more proof methods to cover before finishing Topic 3: proof by contradiction and proof by contrapositive.