Positive Real has Real Square Root

Theorem
Let $x$ be a positive real number.

Then for some real number $y$, $x = y^2$.

That is, $\forall x\in\R^+ \exists y:x\mathop=y^2$

Proof
Let $f\colon \mathbb R \to \mathbb R$,

By Power Function is Continuous, $f$ is continuous.

By Power Function is Unbounded Above, there is some real $q$ such that $f(q) > x$.

$0^2 = 0 \le x$.

By the Intermediate Value Theorem, there is a real $y$ between $0$ and $q$ such that $y^2 = x$.