$\sigma$-algebra of Borel in $\mathbb{R}$ is the power set of $\mathbb{R}$ We define the $\sigma$-algebra of Borel of $\mathbb{R}$ as the $\sigma$-algebra generated by the open intervals $(a,b)$ with $a,b \in \mathbb{R}$, $a<b$.
Let $x\in \mathbb{R}$ and construct an open interval $I=(a,b)$ such that $x\in I$. Then $x \in (a+\frac{n}{n+1}(x-a),b-\frac{n}{n+1}(b-x))$, $\forall n \in \mathbb{N}$.
We have $\{x\} = \cap(a+\frac{n}{n+1}(x-a),b-\frac{n}{n+1}(b-x))$. As any $\sigma$-algebra is closed under countable intersections, $\{x\}$ is contained in our $\sigma$-algebra.
We conclude that this $\sigma$-algebra is the power set of $\mathbb{R}$. However, I haven't seen this fact written in any text, so I wonder if my reasoning is wrong. Can anyone confirm or disprove my statement?

How do you conclude that "this $\sigma$-algebra is the power set of $\mathbb{R}$" ?
You cannot write any element of the power set of $\mathbb{R}$ as a countable union of singletons.
