Prove proposition on real numbers and uniqueness.

Question

How would I go about proving the following proposition. Do I have to prove uniqueness, or that if $x^2 = r$, then $x = \sqrt r$?

Prove given any $r \in \mathbb R\gt 0$, the number $\sqrt r$ is unique in the sense that, if $x$ is a positive real number such that $x^2 = r$, then $ x = \sqrt r$.

Answer 1

If $0 < x < y$ then $x^2 < y^2$. Consequently, if $x^2 = y^2 = r$ and $x,y > 0$, then you must have $x=y$.

Answer 2

Suppose $x > 0, x^2 = r$ and we have $y > 0, y^2 = r$. Then $0 = r - r = x^2-y^2 = (x - y)(x+y)$. But since $x > 0$ and $y > 0$ then $x+y > 0$ so it implies that $x - y = 0$.