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**aurora** I need to use the fact that R is a baire space somewhere, but I just don't seem to be able to prove it. I'm not even sure if it's trivial or not (meaning, if it's even required to prove it), but I guess I can't take the risk. Moreover, it's disturbing I don't know how too.

I know that R is a complete metric space.

I've read that complete spaces are baire spaces, but I haven't got the proof for that.

My definition for baire space is one of the following (which are equivalent, I've proved that):

1) every countable intersection of dense open sets is a dense set.

2) every countable unification of non-dense (not sure of the term) sets has an empty interior.