Exercise 2.3.9: If S C R is a set, then x E R is a cluster point if for every & > 0, the set (x - &,x+ 8)nS \\ {x} is not empty. That is, if there are points of S arbitrarily close to x. For example, S := {1 : n E N} has a unique (only one) cluster point 0, but 0 # S. Prove the following version of the Bolzano-Weierstrass theorem: Theorem. Let S C R be a bounded infinite set, then there exists at least one cluster point of S. Hint: If S is infinite, then S contains a countably infinite subset. That is, there is a sequence {xn} of distinct numbers in S