The following is from A Book of Set Theory by Pinter. Chapter 5, Ex. 5.4.4
5.14 Theorem (Hausdorff’s Maximal Principle). Every partially ordered set has a maximal chain.
5.15 Definition A partially ordered set A is said to be inductive if every chain of A has an upper bound in A.
5.16 Theorem (Zorn’s Lemma). Every inductive set has at least one maximal element
Prove that Zorn’s Lemma is equivalent to the following: Let A be an inductive set and let a ∈ A; then A has at least one maximal element b such that b>=a
Attempted proof: (caveat lector: my proof proof will sound Gobbledygook gook)
Assume the hypothesis: By 5.24 and Def 5.15 let C be a maximal chain in A.Let a be an upper bound in A. Let D be a chain in C
K=$\bigcup D$ for D $\in C$
So K is the l.u.b If a,b$\in $ K then a $\in$ N and b $\in$ M for M,N$\in$ C .By assumption aa so K is totally ordered .So a<=b
By our hypothesis b is maximal . C is a chain,we have the upper bound a $\in $ A Then C $\cup$ {a} is a chain. So by the maximality of C a $\in $ C .
If b>a then C $\cup$ {b} is a totally ordered set. So b$\in$ C; a$\le$ b then a=b. Therefore hypothesis =>ZL is shown.
I hope I got this part correct Any help appreciated. My knowledge of working with ZL is shaky