Borel sigma algebra are generated by open sets Please check my proof:
The Borel sigma algebra $B(\mathbb{R}^d)$ is generated by all (left) half-open intervals. I need to show that it is also generated by all open intervals:
So $(a,b)=\bigcup(a,b-1/n]$
so $(a,b)$ is in the Borel set generated by half open intervals.
Similarly $(a,b]=\bigcup(a,b+1/n)$ hence $(a,b]$ is in the Borel set generated by all open intervals.

I believe you mean $(a,b] = \bigcap (a, b+1/n)$ (rather than $\bigcup$) which is indeed in the $\sigma$-algebra generated by the open intervals. Besides this typo, you're entirely correct!

I hope this helps ^_^
