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proof that every filter is contained in an ultrafilter

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I have never heard of a "subset of X" which is "not contained in X"... this is a very, very careless mistake.

Evidently you're new to the site. Ray (user rspuzio) is one of the PlanetMath's most active contributors; please try not to be so careless yourself and properly follow the site's protocol by filing your correction via the "correct" button at the bottom of the object.

I can do this, but the proof I know is fairly different than this one.

I am now reading the guidelines, sorry for that...

It was a simple misprint, which is now fixed.

In that case, please add your proof. There are certainly different
ways to go about proving the result and having more proofs of the
result presented will improve the coverage of the subject.

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