Set Theory/The axiom of choice

Exercises

 * 1) Prove that Zorn's lemma is equivalent to Tukey's lemma which states that whenever $$X$$ is a set and $$\mathcal F \subseteq 2^X$$ has the property that $$A \in \mathcal F$$ if and only if $$B \in \mathcal F$$ for all finite sets $$B \subseteq A$$, then for all $$Y\subseteq X$$ there exists a maximal $$Z \subseteq X$$ among all sets in $$\mathcal F$$ that contain $$Y$$ (where we consider $$\mathcal F$$ to be ordered by inclusion).