Math 311
Dr. Raynor
Due: Wednesday, April 1 at 11:00am
Test #2: Take-Home Component
You may refer to the written materials from our course for this exam, such as the textbook,
your notes from lecture, and your homework (including any comments I made). You may
not use any other written or recorded materials, including other textbooks. You may not
use the internet. You may not speak with any person about this exam (except me).
This test is out of 70 points, of which 20 points are on this take-home component. It is
due at 11:00am on Wednesday, April 1. You must sign the honor pledge and turn it in
with your exam.
Pledge: I pledge on my honor that I have neither given nor received any assistance on
this exam, nor have I used any dishonest means to obtain my results. I have not referred to
any forbidden materials in completing this exam.
Signature:
(1) (8 points) Prove that if (sn ) and (tn ) are bounded sequences of nonnegative numbers,
then
lim sup(sn tn ) ≤
n→∞
lim sup sn
n→∞
lim sup tn .
n→∞
Does equality necessarily hold? Justify your response.
(2) (10 points) Consider Thomae’s function, which is defined by
(
1
x = pq in lowest terms
t(x) = q
.
0 x 6∈ Q
(a) (5 points) Let x ∈ R and consider a sequence (xn ) of rational numbers so that
xn → x but xn 6= x ∀n ∈ N. Suppose that xn = pqnn . Show that qn → ∞ as
n → ∞.
(b) (7 points) Assuming part (a), prove that t(x) is continuous at every irrational
number and discontinuous at every rational number.
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Math 311
Dr. Raynor
Monday, March 30
Test #2: In-Class Component
NAME:
Pledge: I pledge on my honor that I have neither given nor received any assistance on
this exam nor have I used any dishonest means to obtain my results.
Signature:
Note: This test is out of 70 points, of which 50 points are on this in-class
portion. To receive full credit you must SHOW ALL WORK and JUSTIFY
ALL CLAIMS. Throughout this test, you may assume any theorems proved in the text or
during lecture, unless specifically stated otherwise.
Pledge: I pledge on my honor that I have neither given nor received any assistance on
this exam, nor have I used any dishonest means to obtain my results. I have not referred to
any forbidden materials in completing this exam.
Signature:
(1) True/False: 2 points each. Give a one sentence justification for your answer in each
part.
(a) Every bounded sequence of real numbers converges.
(b) Continuous functions take open intervals to open intervals.
(c) An open set can have an isolated point.
(d) Convergent sequences must be Cauchy.
(e) The union of countably many closed sets must be closed.
(2) (2 points each) State the following definitions and theorems:
(a) A function f is continuous at a point x0 ∈ dom(f ) if...
(b) A sequence (sn ) of real numbers is Cauchy if...
(c) The extreme value theorem says...
(d) A set S is compact if...
(e) A subsequence of a sequence (sn ) is...
(3) (5 points) Prove that every closed, bounded subset of R is compact.
(4) (10 points) State and prove the Bolzano-Weierstrass Theorem.
(5) (7 points) Consider the function
( 2
x −1
x 6= 1
f (x) = x−1
.
a
x=1
Is it possible to choose a real number a so that f will be continuous at 1? If so,
choose the correct value of a and prove that f is continuous at 1. If not, prove that
it is impossible.
P
(6) (8 points)
Prove
that
if
xn converges absolutely, and (yn ) is P
a bounded sequence,
P
then
xn yn converges absolutely. Give a counterexample if
xn only converges
conditionally.
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