We say that a number C is a cluster point of a sequence {xk} if any interval (C − , C + ) contains infinitely many sequence elements.

(a) Explain the difference between a limit and a cluster point.

(b) Provide an example of a sequence with more than one cluster point.

(c) We will prove that every bounded sequence has at least one cluster point, whether it converges or not. Prove the following steps: 

i. Suppose we have a sequence {xk} where every element is in the interval [L, R]. Show that you can construct an interval [L2, R2] ⊆ [L, R] that is half as big and contains infinitely many sequence elements. 

ii. Show that you can therefore construct a shrinking sequence of nested intervals, each of which contains infinitely many sequence elements. 

iii. There is one element C inside all of these intervals. Show that any open interval around C contains one of these intervals, and therefore infinitely many elements. Therefore C is a cluster point.