# Borel signma fields

Denote $\mathcal C$ the collection of $(a,b)$, $-\infty, $\mathcal C_1$ the collection of $[a,b]$, $-\infty, etc... Since $[a,b]=\bigcap_{n\geq 1}\left(a-\frac 1n,b+\frac 1n\right)$, each element of $\mathcal C_1$ is in the $\sigma$-algebra generated by $\mathcal C$, so $\mathcal B_1(\mathbb R)\subset\mathcal B(\mathbb R)$. Use arguments of this type to get the other inclusions.