Hausdorff’s maximality theorem (zorn’s lemma)

Hausdorff’s maximality theorem (or Zorn’s lemma) is used often in graduate level real analysis course. Because the proof is a bit convoluted, most courses use it without proving it and professors usually direct us to the proof in Rudin’s real and complex analysis. There is a two page appendix that proves the Hausdorff’s maximality theorem. Just like most of the proofs in the book, the proof is very dense and it took me a long time to understand it. Below I provide an easy to follow proof in multiple steps.