Results
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. $\square$
