ASSIGNMENT 13
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. The second part consists of the questions on this page. You are expected to provide full solutions
with complete arguments and justifications. You will be graded on the correctness, clarity and elegance of
your solutions. Your answers must be typed or very neatly written. They must be stapled, with your name
and student number at the top of each page.
This assignment is due at 10:00 a.m. on Monday, March 10. The written assignment is to be handed
in by that time. The online assignment will also close at that time.
1. In this question you will prove a result called the Bolzano-Weierstrass theorem.
A subsequence is formed from a sequence by selecting terms from that sequence. More precisely, a
subsequence of a sequence {an } is a sequence of the form {anr }, where the indices nr are strictly increasing.
Suppose the sequence {an } is bounded. Explain how you may construct a convergent subsequence. (Hint:
use the technique of bisection we used to prove the Intermediate Value Theorem.)
2. Consider the fractal curve which is constructed using the following infinite algorithm.
(a) Begin with an equilateral triangle of area 1.
(b) Trisect each line segment. Then make the centre segment the base of an equilateral triangle pointing
outward. Finally, remove the centre segment.
(c) Return to the beginning of step (b).
Find the length of, and area enclosed by, this curve.
3. We proved in class that the harmonic series diverges. In this question we consider an “almost-harmonic”
series.
Let {an } be the sequence defined as follows:
an =
Prove that
X
1
n
0
if n does not contain the digit 9
.
otherwise
an converges.
n≥1
Bonus: Let {bn } be the sequence defined as follows:
bn =
Explain why
1
n
0
X
n≥1
if n does not contain the string of digits given by the first million digits of π
.
otherwise
bn converges.