Arithmetic Mean is Never Less than Harmonic Mean

Theorem
Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers.

Let $A_n $ be the arithmetic mean of $x_1, x_2, \ldots, x_n$.

Let $H_n$ be the harmonic mean of $x_1, x_2, \ldots, x_n$.

Then $A_n \ge H_n$.

Proof
$A_n$ is defined as:
 * $\ds A_n = \frac 1 n \paren {\sum_{k \mathop = 1}^n x_k}$

$H_n$ is defined as:
 * $\ds \frac 1 H_n = \frac 1 n \paren {\sum_{k \mathop = 1}^n \frac 1 {x_k} }$

We have that:
 * $\forall k \in \closedint 1 n: x_k > 0$

From Positive Real has Real Square Root, we can express each $x_k$ as a square:
 * $\forall k \in \closedint 1 n: x_k = y_k^2$

without affecting the result.

Thus we have:


 * $\ds A_n = \frac 1 n \paren {\sum_{k \mathop = 1}^n y_k^2}$


 * $\ds \frac 1 {H_n} = \frac 1 n \paren {\sum_{k \mathop = 1}^n \frac 1 {y_k^2} }$

Multiplying $A_n$ by $\dfrac 1 {H_n}$:

So:
 * $\dfrac {A_n} {H_n} \ge 1$

and so from Real Number Axioms: $\R \text O 2$: compatible with multiplication:
 * $A_n \ge H_n$

Also see

 * Cauchy's Mean Theorem