Q:

How to prove that there is a unique positive real number $x$ such that $x^2 = 2$?

How to prove that there is a unique positive real number $x$ such that $x^2 = 2$?
My step: suppose there are $2$ positive real number $x, y$ such that $x^2=y^2=2$, then we take sqrt of both and we get $\sqrt{x^2}=x=y=\sqrt{y^2}$. Is it right?

A:

Better would be: $$x^2=y^2\implies x^2-y^2=0\implies (x-y)(x+y)=0\implies x=\pm y$$Then if $x=-y$, and both are positive, then $x=y=0$, contradiction. So $x=y$. Also, your proof only deals with the uniqueness aspect of the proof, while the question asks for both uniqueness, and existence. Though perhaps you've done this already.

