Monotone Class Definition:
We say that $\mathcal{G}$ is a monotone class if whenever $\{A_{k}\}$ is an increasing and $\{B_{k}\}$ is a decreasing sequence in $\mathcal{G},$ then $\cup A_{k}$ and $\cap B_{k}$ are in $\mathcal{G},$ as well.
My question is:
How can I prove rigorously that this set $$\{(-\infty, a_{n}), (-\infty, a_{n}], \emptyset, [a_{n}, \infty),\mathbb{R}, (a_{n}, \infty)\}, $$
with $(-\infty, a_n)$ such that $(a_n)$ decreasing, $(-\infty, a_n]$ s.t. $(a_n)$ decreasing, $(a_n,\infty) $ s.t. $(a_n)$ increasing, or $[ a_n, \infty)$ s.t. $(a_n)$ increasing.
My trial:
According to intuition, the countable union of increasing unbounded intervals is in my set, it is just $(a_{1}, \infty)$ or $[a_{1}, \infty)$, but still I donot know how to prove it rigorously, so any help in this step will be appreciated.
for the infinite intersection I do not have any intuition and I am unable to prove it rigorously. so any help in this step will also be appreciated.
is a monotone class?