Let {xn} be a sequence in R with xn → a. Use the definition of a compact set to prove the set {xn : n ∈ N} ∪ {a} is compact

The Cantor set, named after the German mathematician Georg Cantor (1845-1918), is constructed as follows: Start with the closed interval [0, 1] and remove the open interval .  That leaves the two intervals  and . Then remove the open middle third of each of those intervals.  Four intervals remain, and again remove the open middle third of each of them.  Continue this procedure indefinitely, at each step removing the open middle third of every interval that remains from the preceding step.  The Cantor set consists of the numbers that remain in [0, 1] after all those intervals have been removed.   Show that the total length of all the intervals that are removed is 1. Despite that, the Cantor set contains infinitely many numbers. Give examples of at least three numbers in the Cantor set.

For a set A, let P(A) be the set of all subsets of A. Prove that A is not equivalent to P(A). (Hint: Suppose f : A -> P(A) and define C = {x: x ∈ A and x  /∈ f(x)}. Show C /∈ im f .)