Let E be a nonempty subset of Rn.
a) Show that a sequence xk ∈ E converges to some point a ∈ E if and only if for every set U, which is relatively open in E and contains a, there is an N ∈ N such that xk; ∈ U for k > N.
b) Prove that a set C ⊂ E is relatively closed in E if and only if the limit of every sequence xk ∈ C which converges to a point in E satisfies limk→∞ xk ∈ C.