By definition, we have
\begin{equation}
Z \overset{d}{=} \lim_{N \to \infty} \sum_{n = 1}^N \frac{\xi_n}{\sqrt{N}}
\end{equation}
where $\xi_1, \xi_2, \xi_3, \ldots$ are random integers with
\begin{equation}
\mathbb{P}(\xi_n = -1) = \mathbb{P}(\xi_n = 1) = \frac{1}{2}
\end{equation}
for all positive integers $n \geq 1$. Using result
R2405: Characteristic function uniquely identifies the distribution of a random real number, it is thus sufficient to show that the characteristic function of $\lim_{N \to \infty} \sum_{n = 1}^N \frac{\xi_n}{\sqrt{N}}$ is given by $t \mapsto e^{- t^2 / 2}$.
To this end, denote $S_N : = \sum_{n = 1}^N \xi_n$. Results
imply that $N^{-1/2} S_N$ has characteristic function
\begin{equation}
\mathbb{E} \left( e^{i \frac{t}{\sqrt{N}} S_N} \right)
= \left( \cos \frac{t}{\sqrt{N}} \right)^N
\end{equation}
By definition of
D1927: Standard real cosine function, we have
\begin{equation}
\cos x
= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots
\end{equation}
Thus
\begin{equation}
\cos \frac{t}{\sqrt{N}}
= 1 - \frac{t^2}{2! N} + \frac{t^4}{4! N^2} - \frac{t^6}{6! N^3} + \dots
\end{equation}
so that we have the approximation
\begin{equation}
\cos \frac{t}{\sqrt{N}} = 1 - \frac{t^2}{2 N} + o_{t \to 0}(1)
\end{equation}
Therefore, result
R3304: Approximating sequence for the natural exponential function shows that the limit for $(\cos \frac{t}{\sqrt{N}})^N$ exists as $N \to \infty$ and
\begin{equation}
\begin{split}
\mathbb{E} \left( e^{i t Z} \right)
= \mathbb{E} \left( e^{i t \lim_{N \to \infty} N^{-1/2} S_N } \right)
& = \mathbb{E} \left( \lim_{N \to \infty} e^{i \frac{t}{\sqrt{N}} S_N} \right) \\
& = \lim_{N \to \infty} \mathbb{E} \left( e^{i \frac{t}{\sqrt{N}} S_N} \right) \\
& = \lim_{N \to \infty} \left( \cos \frac{t}{\sqrt{N}} \right)^N \\
& = \lim_{N \to \infty} \left( 1 - \frac{t^2}{2 N} + o(1) \right)^N \\
& = \lim_{N \to \infty} \left( 1 + \frac{- t^2 / 2}{N} + o(1) \right)^N
= e^{-t^2 / 2}
\end{split}
\end{equation}
$\square$
