Every open set in R is a countable union of open intervals. Prove.

[seeker101]

I was trying to prove that the sigma algebra generated by the set of open intervals is the same as the sigma algebra generated by the set of open sets. This proof devolves into proving the statement in the title. I think rational numbers must be brought into the picture to prove this stmt but I can't think of how to do it... Any suggestions?

 

[Landau]

See here, here, or here (Theorem 4.1.1) for three different methods.