Given any $r \in \mathbb{R}_{>0}$, the number $\sqrt{r}$ is unique in the sense that, if $x$ is a positive real number such that $x^2 = r$, then $x = \sqrt{r}$
I would appreciate any nudge in the right direction. My initial thought would be to prove that this is impossible for $x<\sqrt{r}$ and $x>\sqrt{r}$.