Show that a set E in the plane is closed if and only if for every convergent sequence (Pro...

Show that a set E in the plane is closed if and only if for every convergent sequence (Problem 1) of points {Pn} in E, the limit of the sequence is in E. [Hint: Suppose E is closed and Pn → P0, with Pn in E for all n. If P0 is not in E, then choose ϵ > 0 such that d(P, P0) < ϵ implies that P is not in E (why is this possible?) and obtain a contradiction. Next suppose E is such that whenever {Pn} is in E and Pn → P0, then P0 is in E. To show that E is closed, suppose that P0 is a point not in E and that no neighborhood of Pq consists solely of points not in E. Then choose Pn such that Pn is in E and d(Pn, P0) < 1 /n. Show that Pn → P0 and obtain a contradiction.]

Problem 1

An (infinite) sequence of points P1, ..., Pn, ... in the plane is said to converge and have limit P0:

if for each ϵ > 0 there is an integer N such that d(Pn, P0) < ϵ for n ≥ N. Show that the limit P0 is unique. [Hint: If and obtain a contradiction.]