Simple Proof Methods

A theorem is a mathematical statement that has already been proven or can be proven to be true based on definitions or previously proved theorems.

Examples:

Theorem. [Sum of even integers] If $m$ and $n$ are both even integers, then their sum, $m + n$, is also even.

Theorem. [Infinity of $\mathbb{Z}$] There exist infinitely-many integers. That is, $|\mathbb{Z}| = \infty$.

Theorem. [Solutions of a quadratic equation] There exist at most $2$ real solutions to a quadratic equation of the form $ax^2 + bx + c = 0$, where $a, b, c \in \mathbb{R}$ and $a \neq 0$.

In $\LaTeX$, theorems are written inside the

theorem

environment; see slide 36 for the full $\LaTeX$ code.