We can write
\begin{equation}
\begin{split}
\sum_{n = 1}^N \frac{X_n - \mu}{\sqrt{\sigma^2 N}}
& = \frac{1}{\sigma} \frac{1}{\sqrt{N}} \left( \sum_{n = 1}^N X_n - \sum_{n = 1}^N \mu \right) \\
& = \frac{1}{\sigma} \frac{\sqrt{N}}{N} \left( \sum_{n = 1}^N X_n - \sum_{n = 1}^N \mu \right) \\
& = \sqrt{N} \left( \frac{\overline{X}_N - \mu}{\sigma} \right) \\
\end{split}
\end{equation}
Hence, this result is a reformulation of
R3843: I.I.D. real central limit theorem with the identity index sequence. $\square$
