When we prove $\sigma((a,b)) = \sigma((a,b])$. All we have to show is that $(a,b]$ can be represented by a countable intersection of $(a,b+1/n)$. And the similar job is done for the other way.

I am not sure why this is enough to show that two sigma algebra is equivalent.
I hope someone can explain, and my explanation is clear enough. Thanks,