Determine, with justification, if the interval $(0,1]$ $\in\mathfrak B(\mathbb R)$ or not, given $\mathfrak B(\mathbb R)$ is the Borel $\sigma$-algebra on $\mathbb R$, which contains all the the open sets.
I'm not really sure how to approach this problem. I'd really appreciate any tips on how to prove this or any solutions.
You just need to remember that countable intersections of Borel sets are Borel, and notice we can write $$(0,1]=\bigcap_{n=1}^{\infty} (0, 1+\frac{1}{n})$$