I'm really quite distraught on two particular questions... I have attempted them, but I am new at proofs and help / ideas as to how to start these two proofs would be greatly appreciated! Thank you in advance!!

Prove that the only set in R1 which is open and closed are the empty set and R1 itself.

Given a set S in Rn with the property that for any x that is an element of Sthere is an n-ball B(x) such that B(x)S is countable. Prove that S is countable.