1. Prove, without applying Theorem 1.5.1, that the well-ordering principle implies AC. Caution: Is it clear that the possibly very large family of sets A(x ) can all be well-ordered simultaneously, so that there is a function f on I such that for each x ∈ I, f (x ) is a well-ordering of A(x )? Hint: Use AC,.
2. Prove it is equivalent to AC that in every partially ordered set, every chain is included in a maximal chain.