Let $p > 0$ be a positive real number. Applying result
R4671: First-degree polynomial approximation for a standard basic real exponential function near zero, we have
\begin{equation}
\frac{1}{N} \sum_{n = 1}^N x_n^p
= \frac{1}{N} \sum_{n = 1}^N (1 + p \log x_n + o_{p \to 0}(p))
= 1 + \frac{p}{N} \sum_{n = 1}^N \log x_n + o_{p \to 0}(p)
\end{equation}
Raising both sides to power $1/p$, this leads to
\begin{equation}
\left( \frac{1}{N} \sum_{n = 1}^N x_n^p \right)^{1/p}
= \left( 1 + \frac{p}{N} \sum_{n = 1}^N \log x_n + o_{p \to 0}(p) \right)^{1/p}
\end{equation}
Result
R2687: Approximating function for the natural exponential function now states that the limit of the right-hand side expression exists as $p \to 0$, in which case
\begin{equation}
\begin{split}
\lim_{p \searrow 0} \left( \frac{1}{N} \sum_{n = 1}^N x_n^p \right)^{1/p}
& = \lim_{p \searrow 0} \left( 1 + \frac{p}{N} \sum_{n = 1}^N \log x_n + o_{p \to 0}(p) \right)^{1/p} \\
& = \exp \left( \frac{1}{N} \sum_{n = 1}^N \log x_n \right) \\
& = \exp \left( \frac{1}{N} \log \prod_{n = 1}^N x_n \right) \\
& = \left( \prod_{n = 1}^N x_n \right)^{1/N} \\
\end{split}
\end{equation}
$\square$
