Set Theory/The axiom of choice

Definition (axiom of countable finite choice):

The axiom of countable finite choice states that whenever ${\displaystyle (S_{n})_{n\in \mathbb {N} }}$ is a countable family of non-empty sets, then there exists a sequence ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ such that ${\displaystyle \forall n\in \mathbb {N} :x_{n}\in S_{n}}$.

Exercises

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