Let $X_1, X_2, X_3, \dots \in \text{Random}(\mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3, \dots$ is an D3358: I.I.D. random collection
(ii)	$\mathbb{E} X_1 = 0$
(iii)	$\text{Var} X_1 = 1$

Let $f : \{ 1, 2, 3, \ldots \} \to \{ 1, 2, 3, \ldots \}$ be a D5406: Positive integer function such that

(i)	\begin{equation} \lim_{N \to \infty} f(N) = \infty \end{equation}

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. $\square$