a) Let E be a subset of Rn. A point a ∈ Rn is called a cluster point of E if E ∩ Br (a) contains infinitely many points for every r > 0. Prove that a is a cluster point of E if and only if for each r > 0, E ∩ Br(a)\{a} is nonempty.
b) Prove that every bounded infinite subset of Rn has at least one cluster point.