Assume that S is a function with domain w such that S(n) is a subset of S(n^+) for each n in w. (Thus S is an increasing sequence of sets.) Assume that B is a subset of the union of S(n)'s for all n such that for every infinite subset B' of B there is some n for which B' intersect S(n) is infinite. Show that B is a subset of some S(n).
Elements of Set Theory, Enderton H.
Page 158 Question: 25
The Attempt at a Solution
I'm a little stuck on even starting this proof (not to mention from just thinking about it I cant seem to reason why it should be true).
I know I need to use the axiom of choice (Its in the axiom of choice chapter). I'm leaning toward a proof by contradiction though I dont know how to proceed. Any suggestions on how to start this proof would be greatly appreciated.