Consider a sigma-algebra generated by a rectangle $R:=[a,\infty) \times (b,\infty)$ on $\mathbb{R}^2$, denoted by $\sigma(R)$, I would like to know if this is equal to the product Borel sigma-algera; i.e., $$\sigma(R) = ? =\mathcal{B}_{\mathbb{R}} \otimes \mathcal{B}_{\mathbb{R}}. $$
Below is my naive thinking: I know $\mathcal{B}_{\mathbb{R}^2} = \mathcal{B}_{\mathbb{R}} \otimes \mathcal{B}_{\mathbb{R}}$ and also know that $\mathcal{B}_{\mathbb{R}}$ can be generated by any interval on $\mathbb{R}$. But I do not see clearly how to link all these pieces together. Any suggestion is appreciated.
It is easy to see that sets of the type $[a,c) \times [b,d)$ belong to $\sigma (R)$. Let $U$ be any open set in $\mathbb R^{2}$. If $x \in U$ then there is a "small" rectangle $[a,c) \times [b,d)$ with $a,b,c,d$ rational containing $x$ which is contained in $U$. It follows that $U$ is the union of these rectangles. This proves that every open set belongs to $\sigma (R)$. Hence every Borel set belongs to $\sigma (R)$.