5. Letn beanatural number. Prove that
13+23+-+rt3 =(1+2+-»+n)3.
6. Letnt and n be natural numbers.
a. Prove that the sum. m +n. also is a natural number. (Hint: Fix m and deﬁne S(n)
to be the statement that m + n is a natural number.)
b. Prove that the pr
4.
I! not (1), WE need to prove (2) must he the me:
Since 5 is not eornpact, so there E (1,.) c S with no convergent subsequenoe
or the subsequenoe converges to ii point outside 5, Considering (l)does riot hupe
pen, (2,.) is bounded Then by Theon 2.33, th
Midterm 1
Math 142A, Lecture C
Fall 2016
Time allowed: 50mins
(40 points total)
Name:
PID:
1
1. (3 points) State the Completeness Axiom for R.
2. Carefully state the following denitions:
(a) (4 points) Dene what it means for a sequence cfw_an to converge
MATH 142A WINTER 2013 PRACTICE MIDTERM
Instructions: You may quote the theorems that we proved in class, or that are proved in
the textbook, in your proofs, unless the problem says otherwise. Generally, do not quote the
result of a homework exercise in yo
MATH 142A
Chapter 3 Review
Jonathan Conder
Continuity
1. Which of the following functions are continuous?
(a) f : R R defined by f (x) := |x|
(b) f : R \ cfw_0 R defined by f (x) :=
(c) f : R \ cfw_1 R defined by f (x) :=
1
x
x2 1
x1
(d) f : R R defined b
MATH 142A
Midterm 2 Solutions
Jonathan Conder
1. (a) This is true (in fact f is uniformly continuous): if > 0 and x, y Z satisfy |x y| < 1, then x = y and
hence |f (x) f (y)| = 0 < .
(b) This is true, by the extreme value theorem (take a maximum for f on
Name:
PID:
Math 142A
Midterm Exam 1
February 1, 2008
Turn off and put away your cell phone.
No calculators or any other electronic devices are allowed during this exam.
You may use one page of notes, but no books or other assistance on this exam.
Read eac
Math 142A Midterm Solutions
Jim Agler
February 20, 2015
1. (i) For a set S R give the definition of sup S.
(ii) Prove that sup [ 0, 1) = 1.
Solution.
(i) When it exists, sup S is the unique real number with the following two properties:
sup S is an upper
Math 142A Practice Midterm Solutions
Jim Agler
1. (i) Give the definition for a set S R to be bounded above.
(ii) Prove that N, the set of natural numbers, is not bounded above.
Solution.
(i) S is bounded above if there exists b R such that b is an upper
MATH 142A
Midterm 1 Solutions
Jonathan Conder
1. (a) This is false, because 1 Q.
(b) This is false; for example N has no upper bound.
(c) This is true. Pick s S, and note that inf S s sup S.
(d) This is true (it was a homework problem).
(e) This is true,
MATH 142A
Homework 6 Solutions
Jonathan Conder
Section 3.4
5. The sequence cfw_f n +
1
n
f (n) does not converge to 0, because
f
n+
1
n
f (n) = 3n +
3
1
+
> 3n
n n3
for every index n N.
8. If f : I R is dened by f (x) :=
f
ba
xa ,
then
a+
ba
3n
f
for ev
MATH 142A
Homework 8 Solutions
Jonathan Conder
Section 4.2
1
1
4. If x, y I and x < y then 1 + x < 1 + y and hence f (x) = 1+x > 1+y = f (y). Therefore f is strictly decreasing.
1
1
1
By the quotient rule, f is dierentiable. Moreover 1 = 1+0 > f (x) > 1+1
MATH 142A
Midterm 1 Solutions
Jonathan Conder
1. (a) This is false, because 1 Q.
(b) This is false; for example N has no upper bound.
(c) This is true. Pick s S, and note that inf S s sup S.
(d) This is true (it was a homework problem).
(e) This is true,
MATH 142A
Homework 4 Solutions
Jonathan Conder
Section 2.3
2. (a) Dene sn := n +
(1)n
n
sn+1 sn = (n + 1) +
for each index n. If n N and n 2 then
(1)n
(1)n+1
n
= 1 + (1)n+1
n+1
n
so sn < sn+1 . Since s1 = 0 <
(1)n
3n
1
(b) Dene sn := n2 +
cfw_sn is not m
MATH 142A
Homework 7 Solutions
Jonathan Conder
Section 4.1
1. (a) This is false: the absolute value function is continuous, but not dierentiable at 0.
(b) This is true, by Proposition 4.5.
(c) This is false: the absolute value function is not dierentiable
MATH 142A
Homework 3 Solutions
Jonathan Conder
Section 2.1
1. (a) This is false. For example, the sequence cfw_(1)n = cfw_1, 1, 1, 1, . . . does not converge, but cfw_(1)n )2
is constant (and converges to 1).
(b) This is false. For example the sequences
MATH 142A
Homework 5 Solutions
Jonathan Conder
Section 3.1
3. If x0 < 0 and cfw_xn is a sequence in R which converges to x0 , then there is an index N such that xn < 0 for all
indices n N. It follows that cfw_f (xN +n ) = cfw_x2 +n converges to x2 = f (
MATH 142A
Homework 2 Solutions
Jonathan Conder
Section 1.2
1. (a) This is false. For example (0, 1) contains no integers, by Proposition 1.6.
(b) This is false. For example (1, 0) contains no positive numbers.
(c) This is true; in fact Q \ Z is dense in R
MATH 142A
Chapter 3 Review
Jonathan Conder
Continuity
1. Which of the following functions are continuous?
(a) f : R R dened by f (x) := |x|
(b) f : R \ cfw_0 R dened by f (x) :=
(c) f : R \ cfw_1 R dened by f (x) :=
1
x
x2 1
x1
(d) f : R R dened by
(e) f
MATH 142A
Exam Solutions
Jonathan Conder
1. (a) True, by induction.
(b) True, because the cube root is continuous (Theorem 3.29).
(c) False. For example, every subsequence of cfw_n is unbounded.
(d) False. For example, if f (x) =
(e) True. For example, if
MATH 142A
Midterm 2 Solutions
Jonathan Conder
1. (a) This is true (in fact f is uniformly continuous): if > 0 and x, y Z satisfy |x y| < 1, then x = y and
hence |f (x) f (y)| = 0 < .
(b) This is true, by the extreme value theorem (take a maximum for f on
MATH 142A
Homework 1 Solutions
Jonathan Conder
Section 1.1
2. (a) This is false. For example (0, 1) is bounded above by 1, but given an element x (0, 1) there is a larger
element, such as x+1 .
2
(b) This is true, because 0 is a lower bound for S, and inf
Math 142A Practice Midterm
Jim Agler
1. (i) Give the definition for a set S R to be bounded above.
(ii) Prove that N, the set of natural numbers, is not bounded above.
2. Show that if an a and bn b, then an + bn a + b.
3. Prove directly from the definitio
MATH 142A
Homework 6 Solutions
Jonathan Conder
Section 3.4
5. The sequence cfw_f n +
1
n
f (n) does not converge to 0, because
3
1
1
f (n) = 3n + + 3 > 3n
f n+
n
n n
for every index n N.
8. If f : I R is defined by f (x) :=
ba
xa ,
then
ba
ba
f a+
= 3n
MATH 142A
Homework 7 Solutions
Jonathan Conder
Section 4.1
1. (a) This is false: the absolute value function is continuous, but not differentiable at 0.
(b) This is true, by Proposition 4.5.
(c) This is false: the absolute value function is not differenti
Suggested Solutions to Homework #07 MATH 4310
Kawai
Section 9.1
(I) The partial sum formula for geometric series:
n
Sn = ark
1
on
k=1
= a 1 + r + r2 + r3 + : + rn
1
=a
1 rn
1 r
; r 6=
1:
If jrj < 1; show that fSn g is Cauchy.
That is, 8" > 0; 9N : m; n >
Math 142A Homework Assignment 3 Selected Solutions 2.2.4 Show that the set of irrational numbers fails to be closed.
Solution Consider the sequence cfw_ . Observe that n is an irrational number for n every index n; for, if r = n were rational, then n r w
Suggested Solutions to Homework #06 MATH 4310
Kawai
Section 3.7
(I) Exercise #8a. This is mostly a topology question.
[Under Fitzpatrick continuity criterion, an isolated point cannot be a limit point, and thus,
s
if a sequence converges to an isolated po
Suggested Solutions to Homework #05 MATH 4310
Kawai
Section 3.3
(I) Exercise #4. Use Intermediate Value Thm.
If we draw the [ 1; 1]
y
[ 1; 1] square, then:
1
0.5
the point ( 1; f ( 1) must be ABOVE the dashed line
and the point (1; f (1) must be BELOW the
Math 142A Homework Assignment 4 Selected Solutions 2.4.10 Prove that a sequence cfw_an does not converge to the number a if and only if there is some > 0 and a subsequence cfw_ank such that |ank - a| for every index k. Solution =) Suppose cfw_an does n