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.