Simple Proof Methods

Fun class activities:

1. Prove that the integer $10$ is even.
2. Prove that the integer $99$ is odd.
3. Prove the following theorem:
   Theorem. [Product of even integers] The product of even integers $n_1, n_2, n_3, \dots, n_m$ is also even.
4. Prove the following corollary:
   Corollary. [Difference of even integers] If $m$ and $n$ are both even integers, then their difference, $m - n$, is also even.
5. Prove or disprove: If $m$ and $n$ are both odd integers, then their sum, $m + n$, is also odd.
6. Definition. [Consecutive integers] Two integers $m$ and $n$ are called consecutive if $m = n + 1$ or if $n = m + 1$.
   Prove or disprove: The product of two consecutive integers is even.