Arithmetic Mean of two Real Numbers is Between them

Theorem
Let $a, b \in \R_{\ne 0}$ be non-zero real numbers such that $a < b$.

Let $\map A {a, b}$ denote the narmonic mean of $a$ and $b$.

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

Proof
By definition of arithmetic mean:


 * $\map A {a, b} := \dfrac {a + b} 2$

Thus:

and:

Hence the result.