Given any r belong to positive real number the number sqrr i

Given any r belong to positive real number, the number sqr(r) is unique in the sense that, if x is a positive real number such that x^2=r, then x=sqr(r).

Please help me to prove it.

Solution

Let, sqr(r) not be unique

Let, u,v be square roots of r

Without loss of generality we can assume: u<v

u^2<uv<v^2

But:u^2=v^2=r

So, r<r and r is positive real number

So we have a contradiction

Hence, sqr{r} is unique