Question

I need ideas on how to begin to prove that the only two sets that are both open and closed are R and ∅.

Answer 1

Here's one strategy: show that every open set in the real must be a union of open intervals. Then show that any union of open intervals is not closed unless it's R or the null set.

Answer 2

You can also try to proof this by contradiction. There's a proof like that in the free book "Topology without Tears", which you'll easily find online. See the proof for proposition 3.3.3 there in the latest edition.