How would you show that Borel $\displaystyle \sigma$-algebra on $\displaystyle \mathbb{R}$ is generated by the sets
$\displaystyle (a,\infty)$, a$\displaystyle \in\mathbb{R}$?
Depends on the definition. The usual definition is that it is generated by sets of the form $\displaystyle (a,b)$. You have to show the generators are the same and use Monotone class theorem. One way you have that $\displaystyle (a,\infty)=\bigcup_{n \in \mathbb{N}} (a,n)$ and the other you have for each n$\displaystyle (a,b-1/n]=(a,\infty) \cap (b-1/n,\infty)$ and also that $\displaystyle (a,b)=\bigcup_{n \in \mathbb{N}} (a,b-1/n]$.
If your definition is different, the method is still the same. Just show that the generators are the same (by showing that you can obtain one set of generators from the other and visa versa).