Geometric Mean of two Positive Real Numbers is Between them

Theorem
Let $a, b \in \R$ be real numbers such that $0 < a < b$.

Let $\map G {a, b}$ denote the geometric mean of $a$ and $b$.

Then:
 * $a < \map G {a, b} < b$

Proof
By definition of geometric mean:


 * $\map G {a, b} := \sqrt {a b}$

where $\sqrt {a b}$ specifically denotes the positive square root of $a$ and $b$.

Thus:

and: