QUESTION
The Cantor set, named after the German mathematician Georg Cantor (1845–1918), is constructed as follows. We start with the closed interval [0,1] and remove the open interval ( 1/3 , 2/3 ). That leaves the two intervals [0, 1/3 ] and [ 2/3 , 1] and we remove the open middle third of each. Four intervals remain and again we remove the open middle third of each of them. We 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.
(a) Give examples of three numbers in the Cantor set and briefly explain why this is the case.
(b) Show that the total length of all the intervals that are removed during this process is 1.
Note: Parts (a) and (b) might appear directly in conflict with each other. The original interval [0, 1] has length 1 and we removed sets with length 1 to create the Cantor set. This should imply that the Cantor set has length zero, but part (a) tells us that there are actually an infinitely many numbers in the Cantor set (the bit that was not removed). To reconcile this, we need more advanced mathematics!