  We present an exposition of the *Chain Bounding Lemma*, which is a common
generalization of both Zorn's Lemma and the Bourbaki-Witt fixed point theorem.
The proofs of these results through the use of Chain Bounding are amongst the
simplest ones that we are aware of. As a by-product, we show that for every
poset $P$ and a function $f$ from the powerset of $P$ into $P$, there exists a
maximal well-ordered chain whose family of initial segments is appropriately
closed under $f$.
  We also provide a "computer formalization" of our main results using the Lean
proof assistant.
