I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The following exercise is Exercise 4 on p.38 in Exercises 2B in this book.
Exercise 4
Suppose $\mathcal{S}$ is the smallest $\sigma$-algebra on $\mathbb{R}$ containing $\{(r,n]: r\in\mathbb{Q},n\in\mathbb{Z}\}.$
Prove that $\mathcal{S}$ is the collection of Borel subsets of $\mathbb{R}.$
We use the following result to solve Exercise 4:
Exercise 3
Suppose $\mathcal{S}$ is the smallest $\sigma$-algebra on $\mathbb{R}$ containing $\{(r,s]: r,s\in\mathbb{Q}\}.$
Prove that $\mathcal{S}$ is the collection of Borel subsets of $\mathbb{R}.$
My solution is here:
Let $r,s\in\mathbb{Q}.$
Let $n$ be an element of $\mathbb{Z}$ such that $s\leq n.$
Then, $(r,s]=(r,n]\cap (\infty,s]=(r,n]\cap (\mathbb{R}\setminus (s, n+1]).$
So, $(r,s]\in\mathcal{S}.$
By Exercise 3, the collection of Borel subsets of $\mathbb{R}$ is a subset of $\mathcal{S}.$Let $r\in\mathbb{Q}$ and $n\in\mathbb{Z}.$
Then, $(r,n]\in\{(r,s]: r,s\in\mathbb{Q}\}.$
By Exercise 3, $\mathcal{S}$ is a subset of the collection of Borel subsets of $\mathbb{R}.$
Is my solution ok?