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Do you know any example that demonstrated Zorn's Lemma simple and intuitive? I know some applications of it, but in these proof I find the application of Zorn Lemma not very intuitive.

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    Many of the most straightforward applications of Zorn’s lemma can be seen as slightly disguised applications of the [Teichmüller-Tukey lemma](http://en.wikipedia.org/wiki/Finite_character). I doubt that you can make either very intuitive, but the special case of sets of finite character seems a little easier to recognize and get a feel for than the general case.2012-12-05

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