Problem 1
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 middle third ($\frac{1}{3}$,$\frac{2}{3}$). That leaves
the two closed intervals [0, $\frac{1}{3}$] and [$\frac{2}{3}$, 1], and we remove the open middle third of each. Four inter-
vals remain, and again we 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.
(a) Draw a picture of the process of constructing the Cantor set.
(b) List six numbers in the Cantor set.
(c) How many numbers are in the Cantor set?
(d) What is the sum of the lengths of the intervals removed in the process of creating the Cantor
set? Write this sum as a series, and evaluate it. Does your answer feel strange?
