Let $x_1, \dots, x_N \in (0, \infty)$ each be a D993: Real number such that

(i)	$A$ is a D2454: Real arithmetic mean for $x_1, \dots, x_N$
(ii)	$H$ is a D3306: Real harmonic mean for $\frac{1}{x_1}, \ldots, \frac{1}{x_N}$

Results

(i)	R4095: Real arithmetic expression for real arithmetic mean
(ii)	R4092: Real arithmetic expression for real harmonic mean

provide the expressions \begin{equation} A = \frac{1}{N} \sum_{n = 1}^N x_n , \qquad H = \frac{N}{\sum_{n = 1}^N \frac{1}{1 / x_n}} \end{equation} Thus \begin{equation} \begin{split} H = \frac{N}{\sum_{n = 1}^N \frac{1}{1 / x_n}} = \frac{N}{\sum_{n = 1}^N x_n} = \frac{1}{\frac{1}{N} \sum_{n = 1}^N x_n} = \frac{1}{A} \end{split} \end{equation} Multiplying both sides by $A$ finishes the proof. $\square$