In attempting to learn about measure theory, I have now come to a problem that I've seen previously on StackExchange.
The exercise is to prove that every interval of $\mathbb{R}$ is a Borel set.
Here is a notable post on the problem Show that every interval is a Borel set
I am wondering what alternate proofs exist besides demonstrating that, given every open subset of $\mathbb{R}$ is a Borel set, the intersections $[a,b] = \bigcap_{n=1}^{\infty} (a-\frac{1}{n}, b+\frac{1}{n})$, $(a,b] = \bigcap_{n=1}^{\infty} (a, b+\frac{1}{n})$, and $[a,b) = \bigcap_{n=1}^{\infty} (a-\frac{1}{n}, b)$ capture all intervals of $\mathbb{R}$.
Where may I find proofs showing that $\sigma \mathcal{C} = \mathcal{B}_{\mathbb{R}}$ (where $\sigma \mathcal{C}$ is the smallest $\sigma$-algebra containing the collection of all intervals of $\mathbb{R}$) or proving this in other ways that do not immediately come to mind?
I would appreciate any wisdom / insights into the matter.