Simple Proof Methods

Fun class activities: Prove the following:

1. There exist infinitely-many even integers.
2. The number $\sqrt{5}$ is an irrational number.
3. If $x^2$ is irrational, then $x$ is irrational.
4. If $n^3$ is an odd integer, then $n$ is odd.
5. Consider an $n \in \mathbb{Z}$. It holds that $n = k$ if and only if $k - 1 < n < k + 1$.
6. No odd integer can be expressed as the sum of $2$ even integers.