Fix $\xi \in \mathbb{R}$ and let $N \geq 1$ be a positive integer. Set $S : = \sum_{n = 1}^{f(N)} X_n$ and let $\mathfrak{F}_S$ denote the characteristic function of $S$. Since $X$ is i.i.d. and since the sum $S$ has $f(N)$ summands, result
R3677: Characteristic function for I.I.D. sum of random euclidean real numbers shows that
\begin{equation}
\mathfrak{F}_S (\xi)
= \left( \mathfrak{F}_{X_1} (\xi) \right)^{f(N)}
\end{equation}
By result
R3844: Characteristic function of a scaled random real number we therefore have
\begin{equation}
\mathfrak{F}_{f(N)^{-1/2} S} (\xi)
= \mathfrak{F}_S (f(N)^{-1/2} \xi)
= \left( \mathfrak{F}_{X_1} (f(N)^{-1/2} \xi) \right)^{f(N)}
\end{equation}
By hypothesis, $\mathbb{E} X_1 = 0$ and $\mathbb{E} |X_1|^2 = 1$, so that using
R3678: , one has the Taylor expansion
\begin{equation}
\left( \mathfrak{F}_{X_1} (f(N)^{-1/2} \xi) \right)^{f(N)}
= \left( 1 - \frac{\xi^2}{2 f(N)} + o \left( \frac{\xi^2}{f(N)} \right) \right)^{f(N)}
= \left( 1 + \frac{- \xi^2 / 2}{f(N)} + o \left( \frac{\xi^2}{f(N)} \right) \right)^{f(N)}
\end{equation}
Result
R3304: Approximating sequence for the natural exponential function shows that the RHS converges to $e^{- \xi^2 / 2}$ as $N \to \infty$. We recognize this as the characteristic function of the standard gaussian (see e.g.
R3680: Characteristic function of standard gaussian random real number). We thus have
\begin{equation}
\begin{split}
\lim_{N \to \infty} \mathfrak{F}_{f(N)^{-1/2} S} (\xi)
= e^{- \xi^2 / 2}
= \mathfrak{F}_{\text{Gaussian}(0, 1)} (\xi)
\end{split}
\end{equation}
Having establishes pointwise convergence for the characteristic functions, the claim is now a consequence of
R4395: Pointwise convergence in characteristic distribution iff convergence in distribution. $\square$
