Without the LaTeX server it is difficult to present a good answer.
But here is an outline. I will skip any finite cases.
Suppose that there are two points a & b such that there does not exist two disjoint neighborhoods one containing a the other b. Now there are countable infinite local bases, one at a U_n and one at b V_n. By the hypothesis, V_n intersect U_n is not empty. So using the axiom of choice, you can chose a sequence that must converge to both a & b.
If you have access to a good mathematics library then look for the topology text by Helen F Cullen. She has given a beautiful proof of this theorem.