Q: Let (a_n) be a sequence of reals so that a = lim(a_n) exists as n→∞. Let S = {a_1, a_2, a_3, · · · } ∪ {a}. Prove that any open cover of S has a finite subcover.

I am trying to prove thiswithoutusing the Heine-Borel Theorem (Heine?Borel theorem - Wikipedia, the free encyclopedia), which is why it is giving me a little trouble. I'm guessing the goal is basically to arrive at a conclusion showing that S is compact, without using the fact that S is bounded and closed (because that's the Heine-Borel Thm).

My intuition would be to show that S is sequentially compact, so that any sequence in S has a subsequence converging to a limit point in S? I'm not sure where to go from here though... Any hint/help would be appreciated.