We use the following lemma in the proof below.
We note the following.
$(4) \Rightarrow (1)$: For every $\bb z$ that is a continuity point of $F_{\bb X}$ we construct a function $h_{\bb z}$ such that $\E(h_{\bb z}({\bb X}^{(n)}))\to\E(h_{\bb z}(\bb X))$ implies $F_{{\bb X}^{(n)}}(\bb z)\to F_{\bb X}(\bb z)$. This will prove the claim that ${\bb X}^{(n)}\tood \bb X$. Specifically, we define $h_{\bb z}(\bb x)=1$ if $\bb x\leq \bb z$ (the inequality holds for all components of the two vectors) and $h_{\bb z}(\bb x)=0$ otherwise. Since $\E h_{\bb z}(\bb x)=F_{\bb x}(\bb z)$, the convergence $\E(h_{\bb z}({\bb X}^{(n)}))\to\E(h_{\bb z}(\bb X))$ implies $F_{{\bb X}^{(n)}}(\bb z)\to F_{\bb X}(\bb z)$. The functions $h_{\bb z}(\bb x)$ satisfy the conditions of (4): they are bounded and measurable, and since \[ \lim_{\epsilon\to 0} F_{\bb X}(\bb z+\epsilon\cdot (1,\cdots,1)) - F_{\bb X}(\bb z-\epsilon\cdot(1,\cdots,1)) =0 \] (recall that $\bb z$ is a continuity point of $F_{\bb X}$), the set of points at which $h_{\bb z}$ is discontinuous has probability 0 (under the distribution of $\bb X$).
$(1) \Rightarrow (2)$: We assume ${\bb X}^{(n)}\tood \bb X$ and consider a continuous function $h$ that is zero outside a bounded and closed set $C$. Since $C$ is compact, and continuous functions on a compact set are uniformly continuous (Proposition B.3.5), $h$ is uniformly continuous. Given $\epsilon>0$, we can find $\delta>0$ such that $\|\bb x-\bb y\| < \delta$ implies $\|h(\bb x)-h(\bb y)\| < \epsilon$. Since $C$ is bounded we can obtain a finite partition $Q$ of $C$ into rectangular cells of the form $\{\bb x: \bb a < \bb x\leq \bb b\}$ such that the distance between points within each cell is less than $\delta$. We can also assume, without loss of generality, that the boundaries between the cells have probability zero under $\bb X$ (since there can only be a countable number of regions with positive probability).
Based on this partition, we can define a simple function (function that takes on finitely many values) whose values are constant on the partition cells \begin{align*} h'(\bb x) = \sum_{i=1}^r a_i I_{A_i} (\bb x) \qquad \text{where} \qquad a_i=h\left(\max_{\bb x\in A_i}(x_1),\ldots,\max_{\bb x\in A_i}(x_d)\right). \end{align*} In fact, we can write the function $h'$ in the following form \[h'(\bb x)=\sum_{i=1}^m \beta_i I_{(-\bb\infty,{\bb b}_i]}(\bb x)\] using the method described in the proof of Proposition 8.2.3 (see also Figure 8.1). Note that by construction of the partition, each point ${\bb b}_i$ is a continuity point of $F_{\bb X}$ and therefore ${\bb X}^{(n)}\tood \bb X$ implies \begin{align} \lim_{n\to\infty} \E(h'({\bb X}^{(n)})) = \lim_{n\to\infty} \sum_{i=1}^m \beta_i F_{{\bb X}^{(n)}}(\bb x) = \sum_{i=1}^m \beta_i F_{{\bb X}}(\bb x) = \E(h'({\bb X})). \tag{**} \end{align} We finally note that \begin{align*} &|\E(h({\bb X}^{(n)}))-\E(h(\bb X))| \\ &= |\E(h({\bb X}^{(n)})) +\E(h'({\bb X}^{(n)}))-\E(h'({\bb X}^{(n)}))+\E(h'({\bb X}))-\E(h'({\bb X})) -\E(h(\bb X)) |\\ &\leq |\E(h({\bb X}^{(n)}))-\E(h'({\bb X}^{(n)}))| + |\E(h'({\bb X}^{(n)})) -\E(h'({\bb X}))| + |\E(h'({\bb X}))-\E(h(\bb X)) |\\ &\leq \epsilon + |\E(h'({\bb X}^{(n)})) -\E(h'({\bb X}))| + \epsilon \\ &\to 2\epsilon \qquad \text{ as } n\to\infty. \end{align*} Above, the first inequality follows from the triangle inequality, the second inequality follows from the uniform convergence of $h$ and the construction of the partition $Q$ and $h'$, and the convergence follows from (**). Since this holds for all $\epsilon>0$, we have $|\E(h({\bb X}^{(n)}))-\E(h(\bb X))|\to 0$ or $\E(h({\bb X}^{(n)}))\to\E(h(\bb X))$.
$(2) \Rightarrow (3)$: For an arbitrary continuous function $h$ such that $\|h(\bb x)\|\leq M$ and an arbitrary $\epsilon>0$ we will show that $|\E(h({\bb X}^{(n)}))-\E(h(\bb X))|\to \epsilon$. Since this holds for all $\epsilon>0$, this implies $\E(h({\bb X}^{(n)}))\to\E(h(\bb X))$.
It is possible to find $\alpha$ such that $\P(\|\bb X\|\geq \alpha) < \epsilon/(2M)$, and a continuous function $0\leq h'(\bb x)\leq 1$ such that $h'(\bb x)=0$ if $\|\bb x\|\geq \alpha+1$ and $h'(\bb x)=1$ if $\|\bb x\|\leq \alpha$. It follows that $\E(h'(\bb X))\geq 1-\epsilon/(2M)$ and \begin{align*} &|\E(h({\bb X}^{(n)}))-\E(h(\bb X))| \\ &= |\E(h({\bb X}^{(n)}))-\E(h(\bb X)) + \E(h({\bb X}^{(n)})h'({\bb X}^{(n)})) \\ &\quad -\E(h({\bb X}^{(n)})h'({\bb X}^{(n)}))+ \E(h({\bb X})h'({\bb X})) - \E(h({\bb X})h'({\bb X}))|\\ &\leq |\E(h({\bb X}^{(n)}))-\E(h({\bb X}^{(n)})h'({\bb X}^{(n)}))| +|\E(h({\bb X}^{(n)})h'({\bb X}^{(n)}))-\E(h({\bb X})h'({\bb X}))| \\ &\quad +|\E(h({\bb X})h'({\bb X})) - \E(h(\bb X))| \\ &\to |\E(h({\bb X}^{(n)}))-\E(h({\bb X}^{(n)})h'({\bb X}^{(n)}))| +0 +|\E(h({\bb X})h'({\bb X})) - \E(h(\bb X))|\\ &\to \epsilon/2+\epsilon/2=\epsilon. \end{align*} The first inequality above follows from the triangle inequality. The first convergence follows from statement (2) and the fact that $hh'$ is a continuous function that vanishes outside a bounded and closed set. The second convergence follows from \begin{align*} |\E(h({\bb X}^{(n)}))-\E(h({\bb X}^{(n)})h'({\bb X}^{(n)}))| &\leq \E(|h({\bb X}^{(n)})|\cdot | 1- h'({\bb X}^{(n)})|) \\ &\leq M \E(| 1- h'({\bb X}^{(n)})|) \\ &= M \E(1- h'({\bb X}^{(n)})) \\ &= M (1- \E h'({\bb X}^{(n)})) \\ &\to M (1- \E h'({\bb X})) \\ &\leq M \epsilon/(2M)=\epsilon/2 \end{align*} (with a similar bound applying to the term $|\E(h({\bb X}))-\E(h({\bb X})h'({\bb X}))|$).
$(3) \Rightarrow (4)$: Let $h$ be a bounded measurable function, for which $\P(\bb X\in \{\bb x: h\text{ is continuous at }\bb x\})=1$. Given $\epsilon>0$, we use the previous lemma to obtain continuous and bounded functions $m,M$ such that $m\leq h\leq M$ and \begin{align*} \E(h(\bb X)) - \epsilon &\leq \E m(\bb X) \\ &= \lim \E m({\bb X}^{(n)}) \\ &\leq \liminf \E( h ({\bb X}^{(n)}))\\ &\leq \limsup \E(h({\bb X}^{(n)})) \\ &\leq \lim \E M({\bb X}^{(n)}) \\ &= \E(M(\bb X))\\ &\leq \E(h(\bb X))+\epsilon. \end{align*} Since $\epsilon$ is arbitrary, $\liminf \E(h({\bb X}^{(n)}))=\limsup \E(h({\bb X}^{(n)}))$, and $\E(h({\bb X}^{(n)}))\to \E(h(\bb X))$.