Question

Suppose (L,≤) is a partially ordered set with the property that every countable chain in L has an upper bound, i.e., if a0≤a1≤a2≤… are in L then there is some b∈L such that an≤b for all n. Let S be a countable subset of L such that for all a,b∈S there is some c∈S such that a≤c and b≤c. Prove that S has an upper bound in L.

Answer 1

Perhaps prove that for each \(x \in S\) that \(x\) belongs to a countable chain, otherwise there would be an element in \(S\) with no upper bound. You might be able to do this by contradiction. Perhaps if there was an \(x \in S\) that was not in a countable chain then \(S\) would not be countable, and prove it using the given property of \(S\)