Order is Preserved on Positive Reals by Squaring

Theorem
Let $$x, y \in \mathbb{R}: x > 0, y >0$$.

Then $$x < y \iff x^2 < y^2$$.

Proof

 * First, we assume $$x < y$$.

$$ $$ $$


 * Now we assume $$x^2 < y^2$$.

$$ $$ $$ $$ $$ $$

An alternative approach is to assume that $$x^2 < y^2$$ but $$x \ge y$$ and obtain a contradiction.