My lecture notes gives the following definition of convergence in distribution of $\mathbb{R}^k$-valued random variables ($x \leq y$ for $x,y \in \mathbb{R}^k \Longleftrightarrow x_i \leq y_i$ for all $i \in \{1,\cdots,k\}$).
Definition 1.6. (Convergence in Distribution): Let $\left\{X_n: n \geq 1\right\}$ and $X$ be random vectors on $\mathbb{R}^k . X_n$ is said to converge in distribution to $X$, denoted $X_n \stackrel{\mathrm{d}}{\rightarrow} X$ if, for all $x$ for which $\mathbb{P}(X \leq x)$ is continuous, as $n \rightarrow \infty$, $$ \mathbb{P}\left(X_n \leq x\right) \rightarrow \mathbb{P}(X \leq x) $$
And then states, as part of the Portmanteau Lemma, that the above definition is equivalent to the following more standard definition of distributional convergence:
$\mathbb{P}\left(X_n \in B\right) \quad \rightarrow \quad \mathbb{P}(X \in B)$ for every Borel set $B \in \mathcal{B}$ such that $\mathbb{P}(x \in\{\operatorname{cl}(B) \backslash \operatorname{int}(B)\})=0$.
The proof is not given and I'm not able to see why the first definition implies the second (other than in the case where $k=1$). Also, is this a standard result? If yes, any reference would also be most appreciated. I found the second but not the first definition in most other online lecture notes that I found.