1. is second countable (open balls centered at rational coordinates and having rational radii form a countable base).
2. Every subspace of a second countable space is second countable (the obvious proof works).
3. Every open cover of a second countable space has a countable subcover.
(Pf: Each set in an arbitrary open cover is some union of those basis sets, so the union of all the sets in the cover is the union of all of those associated basis sets. Now for each of those basis sets in that final union, pick one set of the original cover that it's a subset of. That picks out a countable subset of the original cover, since the basis is countable. The union of all those basis sets covers the space, and is also a subset of the union of that countable subset of the original cover. Thus that countable subset of the original cover is also a cover. This has shown how to produce a countable subcover for any open cover.)
With those onbservations in hand, the problem is straighforward to prove.