Let L be a partially ordered set. Every countable chain in L has an upper bound. Let S be a countable subset of L such that for arbitrary a,b in S there exists a c in S such that a (less-than-or-equal) c and b (less-than-or-equal) c. Prove S has an upper bound in L.
Definition of linear, partial orders.
The Attempt at a Solution
I assume the goal is to prove that S is linearly ordered and thus a chain, having an upper bound in L by the problem text. S is clearly a partial order because it is a subset of L. After listing the properties of a partial order, I have no idea how to continue. Any help would be greatly appreciated! Thanks!