Download HW3-Cantor set

Homework 3
Math320-001
Spring, 2017
Problems 5.2, 5.6, 7.2, 7.4, A.1, A.2, A.3 are compulsory, A.4∗ and A.5∗ are bonus problems
5.2 Give the infimum and supremum of
(a) {x ∈ R : x < 0} (b) {x ∈ R : x3 ≤ 8}
(c) {x2 : x ∈ R}
(c) {x ∈ R : x3 < 8}
5.6 Let S and T be nonempty subsets of R such that S ⊆ T . Prove inf T ≤ inf S ≤ sup S ≤ sup T .
Compare Exercise 4.7(a).
7.2 Determine whether the following sequences converge. If it converges, give its limit. No proofs are
required.
1
3n+1
(a) sn = 3n+1
(b) bn = 4n+1
nπ
n
(d) sin 4
(c) cn = 3n
7.4 Give examples of
(a) A sequence (xn ) of irrational numbers having a limit lim xn that is a rational number.
n→∞
(b) A sequence (rn ) of rational numbers having a limit lim rn that is an irrational number.
n→∞
The Cantor set
Let us perform the following procedure: take a closed interval [0, 1] and remove an open interval (1/3, 2/3).
Repeat this procedure of removing the middle open intervals of a one-third lengths (1/9 on the second step)
with the remained intervals [0, 1/3] and [2/3, 1]. We then continue this process ad infinitum (see the picture
for the first four steps). We call the set of points remaining unremoved after infinite number of steps the
Cantor set.
n=1
n=2
n=3
n=4
A.1 Prove by mathematical
induction that the total length Ln of all intervals remaining in the set after
n
the nth step is 23
A.2 What is the limit of Ln as n → ∞?
A.3 Prove that the Cantor set is nowhere dense, that is, every closed interval [a, b] with 0 ≤ a < b ≤ 1,
where a and b belong to the Cantor set, contains an open interval, not belonging to the Cantor set.
[Hint: first prove that we have a point not belonging to the Cantor set and lying inside [a, b]. Use,
e.g., the result of point A.2 about the total length of intervals. Second, due to the described above
procedure of constructing the Cantor set, every point that does not belong to the set was taken away
at a uniquely defined nth step together with a whole open interval containing this point.]
A.4∗ Endpoints of all intervals arising during the construction procedure belong to the Cantor set by construction. Demonstrate that we have other points of the Cantor set. Show that the point 1/4 belongs
to the Cantor set. [Hint: find the ratio of the distance between the point 1/4 and the closest endpoint
of the interval containing this point after the nth step to the length of this interval.]
A.5∗ Actually, the Cantor set is really big! Prove that it is uncountable. [Hint: use the ternary numeral
system based on three digits {0, 1, 2}. In this system every real number between 0 and 1 has the form
0.1202211221002020101100 . . .. For instance, 0.1 is 1/3 ∈ Q, 0.2 is 2/3 ∈ Q, 0.11 is 4/9 ∈ Q, etc. Then
prove by the mathematical induction in the order of ternary expansion that every number containing
1 anywhere in its expansion will be removed at the very first appearance of 1 besides the only case
where it has only zeros after, which corresponds to left endpoints of closed intervals, like 1/3 = 0.1.
But every number 0.xxxxxx1000000 . . . has an equivalent representation 0.xxxxxx0222222 . . . , so we
can conclude that every point in the Cantor set is in one-to-one correspondence with a ternary number
0.22002202000202000202 . . . that contains only 0’s and 2’s. Now turn to a binary system to compare
with the real numbers on the interval [0, 1].]
Additional information on ternary numeral system and on the Cantor set can be found, e.g., in
https://en.wikipedia.org/wiki/Cantor set
https://en.wikipedia.org/wiki/Ternary numeral system