This should be a pretty routine verification on the compatibility of definitions of product space.
(Product space) Let ${(R_\alpha, {\mathcal B}_\alpha)_{\alpha \in A}}$ be a family of measurable spaces indexed by a (possibly infinite or uncountable) set ${A}$. We define the product ${(\prod_{\alpha \in A} R_\alpha, \prod_{\alpha \in A} {\mathcal B}_\alpha)}$ on the Cartesian product space ${\prod_{\alpha \in A} R_\alpha}$ by defining ${\prod_{\alpha \in A} {\mathcal B}_\alpha}$ to be the ${\sigma}$-algebra generated by the basic cylinder sets of the form
$\displaystyle \{ (x_\alpha)_{\alpha \in A} \in \prod_{\alpha \in A} R_\alpha: x_\beta \in E_\beta \}$
for ${\beta \in A}$ and ${E_\beta \in {\mathcal B}_\beta}$. For instance, given two measurable spaces ${(R_1, {\mathcal B}_1)}$ and ${(R_2, {\mathcal B}_2)}$, the product ${\sigma}$-algebra ${{\mathcal B}_1 \times {\mathcal B}_2}$ is generated by the sets ${E_1 \times R_2}$ and ${R_1 \times E_2}$ for ${E_1 \in {\mathcal B}_1, E_2 \in {\mathcal B}_2}$.
Show that $\mathbb{R}^n$ with the Borel ${\sigma}$-algebra is the product of ${n}$ copies of ${{\mathbb R}}$ with the Borel ${\sigma}$-algebra.
Attempt: We want to show that $\prod_{i=1}^n \mathcal{B}[\mathbb{R}] = \mathcal{B}[\mathbb{R}^n]$(here $\mathcal{B}[\mathbb{R}^d]$ means the Borel $\sigma$-algebra of $\mathbb{R}^d$). By definition, it means that $\displaystyle \sigma(\{(x_1, \ldots, x_n) \in \mathbb{R}^n: x_i \in E_i \in \mathcal{B}[\mathbb{R}], 1 \leq i \leq n\}) = \mathcal{B}[\mathbb{R}^n]$.
First we show that RHS $\subset$ LHS. Since $\mathcal{B}[\mathbb{R}^n]$ can be shown to be generated by the boxes, let $B = I_1 \times \ldots \times I_n$ be a box, where the $I_j$'s are intervals. Then $B = (I_1 \times \mathbb{R} \times \ldots \times \mathbb{R}) \cap (\mathbb{R} \times I_2 \times \mathbb{R} \times \ldots \times \mathbb{R}) \cap \ldots \cap (\mathbb{R} \times \mathbb{R} \times \ldots \times I_n) \in$ LHS. By the definition of a $\sigma$-algebra, this implies that RHS $\subset$ LHS.
Now for the other direction. The generators of the product algebra $E_1 \times \mathbb{R} \times \ldots \times \mathbb{R}, \mathbb{R} \times E_2 \times \ldots \times \mathbb{R}, \ldots ,\mathbb{R} \times \mathbb{R} \times \ldots \times E_n$ are members of $\mathcal{B}[\mathbb{R}^n]$ by the fact that if $E$ and $F$ are Borel sets of $\mathbb{R}^{d_1}$ and $\mathbb{R}^{d_2}$ respectively, then $E \times F$ is a Borel set of $\mathbb{R}^{d_1 + d_2}$. From this, and again by the definition of a $\sigma$-algebra, we also have that LHS $\subset$ RHS.
Are there anything still missing in this argument, or if the argument itself a valid one?