MATH 413 HW 2: Solution to selected problems.
September 12, 2007
2.2.4.3-4 If x is a real number show that there exists a sequence of rational numbers x1 , x2 , . . . representing x such that xn xn+1 and xn < x for all n. Proof: Suppose that n yn i
1
Logic and Sets
Anil Nerode
Propositional Logic
All mathematical statements are built up using logical connectives.
All mathematical statements can be built up
using only the connectives "not", "and", "or", "implies", "i" ,
and the quantiers "there exist
Topics using Uniformity
Math 4130
Anil Nerode
Theorem Let M1 be a metric space,let
M2 be a complete metric space. Suppose that A
M1 is dense in M1,
and that f : A ! M2 is uniformly
continuous.Then there is a unique uniformly continuous g : M1 ! M2
such th
Math 4130 Sample test 1-Nerode
All problems have equal weight.
1a) If fan g is a sequence of reals and b is a real number, dene the phrase
"b is the limit of the sequence fan g "
1b) Prove that if an = 1 31 , then 1 is the limit of the sequence fan g
n
1
Station Eleven Emily St. John Mandel
Reading Journal
1.
Emily St. John Mandel was born and raised on the Denman Island off the west coast of
British Columbia, Canada (she was born in 1979). She studied contemporary dance at The
School of Toronto Dance The
1
Math 4130
Anil Nerode
Continuous Functions
on Metric Spaces
Let (M1;
spaces,
1 ); (M 2 ; 2 )
be metric
let f : (M1; 1) ! (M2; 2): be a
function. Suppose a 2 M1:
2
Denition: f : (M1;
is continuous at
written
1)
! (M 2 ; 2 )
a 2 M1 ;
limx!af (x) = f (a);
Topics using Uniformity
Math 4130
Anil Nerode
Theorem Let M1 be a metric space,let
M2 be a complete metric space.
Suppose that A M1 is dense in M1,
and that f : A ! M2 is uniformly
continuous.
Then there is a unique uniformly continuous g : M1 ! M2 such t
Math4310 Fall2013 Sample Exam 2
The actual test problem will be shorter. I will call upon students next Tuesday to put solutions to all problems on the classroom blackboard for discussion.
1 a) Complete the following denition: "(A; ) is a metric space if.
2, 4, 6b), 6c) Prelim 1 Solutions
October 17, 2013
2a) Dene b is the limit of the sequence cfw_an .
Solution: > 0, N s.t. n > N, |an b| < .
1
2b) Prove that if an = n , then 0 is the limit of the sequence cfw_an .
Solution: Let > 0. Pick N N satisfying N
1
Riemann Integration
Anil Nerode
Math 4130
The foundation of integration as a method for computing areas goes back to
Eudoxus (408-355 B.C.), Euclid (325-265 B.C. ), and
Archimedes (287-212b.c.).
The strategy was to take a bounded plane region
(such as t
1
Math 4130
Anil Nerode
Continuous Functions
on Metric Spaces
Let (M1; 1); (M2; 2) be metric
spaces, let f : (M1; 1) ! (M2; 2):
be a function. Suppose a 2
M1 :
2
Denition: f : (M1;
1)
! (M 2 ; 2 )
is continuous at a 2 M1; written
limx!af (x) = f (a);if an
1
Math 4130
Anil Nerode
Foundations of Calculus
1
Theorem: Suppose f is dened and bounded
on interval [a; b] = [xja x b] with a < b: Suppose f
is Riemann integrable on [a; b] and we dene g(x) =
Rx
f (x)dx on [a; b]:If a < x0 < b and f is continuous at
a
x
1
Ascoli, Arzela, Peano
Anil Nerode
Example:
A limit f (x) of a sequenceffn(x)gof
continuous functions may be discontinuous. Let fn :
[0; 1] ! [0; 1] be dened as fn(x) = xn: The fn
are continuous but f (x) = limn!1 fn(x) is not since
limn!1 fn(x) = 0 for
1
Math 4013- Assignment 2 Due
in class Thursday Sept 12
1) Read module 1 through exercise 10. Work as homework problems 3, 4, 5, 6, 7, 8, 9, 10
2) Read Apostol sections 1.1-1.14. Work as homwork
problems 1.15, 1.16 on P. 26
3) If you have Rosenliicht, rea
1
MATH 4130
Fall 2013
Topics
Anil Nerode
Construction of Reals from Rationals by Cauchy sequences
Alternate constructions of Weierstrass and Dedekind.
Density of Rationals, Archimedian property, completeness
The categoricity of the complete ordered eld ax
Math 4130
ANIL NERODE
Metric Spaces
Let
R
be the set of reals.
A metric space is a non-empty set
together with a function
:M
M
!R
M
such that for all x; y 2 R
1)
(x; y ) = (y; x)
2)
(x; y )
3)
(x; y ) = 0
4)
(x; y ) + (y; z )
0
i x = y
(x; z )
Number 4) a
1
Ascoli, Arzela, Peano
Anil Nerode
Example:
A limit f (x) of a sequenceffn(x)gof
continuous functions
may be discontinuous. Let
dened as fn(x) = xn:
fn : [0; 1]
! [0; 1] be
The fn are continuous but f (x) = limn!1 fn(x) is
not
since limn!1 fn(x) = 0 for
Real Numbers
Anil Nerode
Math 4130
Preface
I try to present mathematical ideas, intuitions,
denitions, and proofs in the historical order in
which they were discovered, or, at a minimum,to
explain how they were discovered. My objective is to demystify mat
Riemann Integration
Anil Nerode
Math 4130
The foundation of integration as a method for computing areas goes back to Eudoxus (408-355 B.C.), Euclid
(325-265 B.C. ), and Archimedes (287-212b.c.). . The
strategy was to take a bounded plane region (such as t
1
Math 4130
Anil Nerode
Elements of Series
Given an innite sequence a0; a1; :of
real numbers, the innite sequence
Pn= a0+: + an of partial sums of the
rst n elements of the sequence
denes the innite series a0+: + an+:
.The innite series a0+: + an+:+ is
ca
1
Functions, Maps,
Equivalence Relations
Anil Nerode
The notions of function, map, and equivalence relation
are ubiquitous in mathematics.
A clear understanding of these notions is fundamental for
reading and understanding
almost all contemporary mathemat
1a)
D"firn " cfw_o^l is a C,r-.\ ,7*^*^
.
;f fo, everd pos it'*e ,nh"nol
ft"L't erists N n N s' * ' f" a'll nt4 Srale ' tl',*, N
cfw_a,r
;s a G*b
s-e1*mr<-
lo^-r.nl 1e
1b)
is; c"^*'l'a
U",t la (eLc +l^*t [t/.,t , +'"l
*1*.
.Flx lot ^/
n
v,fl =J
cfw_
n-tu
MATH 4130 Variation of Sample Test 1
By: Boom!
Instructions: The test is worth a total of 150 points. All problems (a), (b), (c) within a
question have equal weight at 10 points each. Partial credit can be 5 points or 0 points, and
nothing else. You have
1
Logic and Sets
Anil Nerode
Propositional Logic
All mathematical statements are built up using logical
connectives.
All mathematical statements can be built up
using only the connectives "not", "and", "or", "implies",
"i" ,
and the quantiers "there exist
Sur une généralisation de lintégrale dénie
On a generalization of the denite integralz
Note by Mr. H. Lebesgue. Presented by M. Picard.
In the case of continuous functions, the notions of the integral and antideriva-
tives are identical. Riemann dened the
MATH 4130 Quiz Yourself on Prelim 1 Definitions!
1. Limit
A sequence of real (or rational) numbers has limit L if :
2. Cauchy Sequence
A sequence
is Cauchy if:
3. Positive Cauchy Sequence
A Cauchy sequence of rational numbers
is defined to be positive if:
MATH 4130 Extra Exercises for Prelim 1
1. Prove that if
2. (a)
(b)
is a sequence of real numbers that has a limit, then
is Cauchy.
Prove that the sum
of Cauchy sequences of rational numbers
is also a Cauchy sequence.
Prove that the product
of Cauchy seque