1. ## Help with Analysis please :)

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 S there is an n-ball B(x) such that B(x)S is countable. Prove that S is countable.

2. Originally Posted by Majialin
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.
There are different ways of defining "open" and "closed" set. The correct way of proving this depends on what definitions you are using.

One definition is that a set is "open" if it contains none of its "boundary" points and "closed if it contains all of its "boundary" points. If that is the definition you have then being "open" and "closed" means that "all" and "none" of its boundary points must be the same. What are the boundary points of the empty set? What about R? Can you show that any other set must have at least one boundary point?

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