Simple Proof Methods

Consider the following proof by cases:

Theorem. For any real number $x$, it holds that $|x|^2 = x^2$.

Proof. By the definition of absolute value, we consider two cases:

• Case 1: If $x \ge 0$, then $|x| = x$, so: $|x|^2 = x^2$.
• Case 2: If $x < 0$, then $|x| = -x$, so: $|x|^2 = (-x)^2 = x^2$.

In both cases, we obtain $|x|^2 = x^2$, proving the theorem.◼

The reason that we needed to look into these 2 separate cases is that the absolute value behaves differently for different values of $x$, so we needed to verify the equation for all cases to make sure that the equation is true for any integer $x$.