The proof reduces to the univariate CLT via the Cramér–Wold device: a sequence of random vectors $Z_n$ in $\mathbb{R}^k$ converges in distribution to $Z$ if and only if $\alpha^\top Z_n \xrightarrow{d} \alpha^\top Z$ for every fixed $\alpha \in \mathbb{R}^k$. For each $\alpha$, the scalar sequence $\alpha^\top X_1, \ldots, \alpha^\top X_n$ is i.i.d. with mean $\alpha^\top \mathbb{E}[X]$ and variance $\alpha^\top \Sigma \alpha < \infty$. The univariate CLT gives $\sqrt{n}\,\alpha^\top(\bar{X}_n - \mathbb{E}[X]) \xrightarrow{d} \mathcal{N}(0, \alpha^\top \Sigma \alpha)$. Since this holds for all $\alpha$, the Cramér–Wold device gives $\sqrt{n}(\bar{X}_n - \mathbb{E}[X]) \xrightarrow{d} \mathcal{N}(0,\Sigma)$.