Compactness Theorem and partial orderings

Hello folks,

I've been asked 'If P is a partial ordering, how do I use the compactness theorem to show that P is the union of k chains iff each finite subset on P is the union of k chains?'

But, I have absolutely no idea what this question is even driving at. Set theory and logic is easily my weakest area of maths and any help would be much appreciated (I will attempt to reciprocate in differenctial equations or algebra - things I can actually do!)

Thanks in advance.