Checking an example of a monotone class.
Solution 1:
Your example is correct except that you forgot to include $\mathbb R$. The only way you get a decreasing sequence in this class is by considering $(-\infty, a_n)$ with $(a_n)$ decreasing, $(-\infty, a_n]$ with $(a_n)$ decreasing or $(a_n,\infty) $ with $(a_n)$ increasing, or $[ a_n, \infty)$ with $(a_n)$ increasing. Similarly we can write down increasing sequences.
It is not a sigma algebra because $\{0\}=(-\infty,0]\cap [0,\infty)$ does not belong to it.