Harmonic 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 $\map H {a, b}$ denote the narmonic mean of $a$ and $b$.

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

Proof
By definition of harmonic mean:


 * $\dfrac 1 {\map H {a, b} } := \dfrac 1 2 \paren {\dfrac 1 a + \dfrac 1 b}$

Thus:

But $\dfrac 1 {\map H {a, b} }$ is the arithmetic mean of $\dfrac 1 b$ and $\dfrac 1 a$.

Hence from Arithmetic Mean of two Real Numbers is Between them:
 * $\dfrac 1 b < \dfrac 1 {\map H {a, b} } < \dfrac 1 a$

So by Reciprocal Function is Strictly Decreasing:


 * $b > \map H {a, b} > a$

Hence the result.