[proofplan] A direct reduction to the standardised [Central Limit Theorem](/theorems/521) via the substitution $Y_k = (X_k - \mu)/\sigma$. [/proofplan] [step:Standardise the variables] Define $Y_k = (X_k - \mu)/\sigma$ for $k \geq 1$. Then the $Y_k$ are i.i.d. with $\mathbb{E}[Y_1] = 0$ and $\mathbb{E}[Y_1^2] = \operatorname{Var}(X_1)/\sigma^2 = 1$. [/step] [step:Apply the standardised CLT to $(Y_k)$] By the [Central Limit Theorem](/theorems/521), \begin{align*} \frac{Y_1 + \cdots + Y_n}{\sqrt{n}} \xrightarrow{d} N(0,1). \end{align*} [/step] [step:Translate back to obtain the general-variance statement] Substituting $Y_k = (X_k - \mu)/\sigma$: \begin{align*} \frac{Y_1 + \cdots + Y_n}{\sqrt{n}} = \frac{1}{\sqrt{n}}\sum_{k=1}^n \frac{X_k - \mu}{\sigma} = \frac{S_n - n\mu}{\sigma\sqrt{n}}. \end{align*} Therefore $\frac{S_n - n\mu}{\sigma\sqrt{n}} \xrightarrow{d} N(0,1)$. [/step]