Math 415 Notes and Summaries August 2008: This page requires Mozilla/Firefox/Netscape 7+ or Internet Explorer 6+ MathPlayer .

Martin Flashman's Courses
MATH 415 Real Analysis I Fall, 2008
Class Notes and Summaries
 Week1: Beginnings. 8-25 8-28 week 2 9-1 No class 9-4 week 3 9-8 9-11 week 4 9-15 9-18 week 5 9-22 9-25 week 6 (5+) 9-26(Fri) 10-2 week 7 (6+) 10-3 (Fri) 10-6  (Mon.) week 8 10-13 10-16 week 9 10-20 10-23 week 10 10-27 10-30 week 11 11-3 11-6 week 12 11-10 11-13 week 13 11-17 11-20 week 14 NoClasses week 15 12-1 12-4 week 16 12-8

Week 1:
• Other texts in Real Analysis:

• HSU Library Holdings
• On Proofs:

• How to Read and Do Proofs by D.Solow
The Keys to Advanced Mathematics by D. Solow
How to Solve It by G. Polya

8-28
• We watched the film  "Space Filling Curves": in the library on shelf- please replace accurately! VIDEO2577
• Use of point-wise convergence for a sequence of approximating curves,
• Use of concept of continuity for curves.
• Length of a curve.
• How does a curve "fill" space?
• In respnse to a question about how to make sense of curves being close and sequence of curves having limits, we introduced the concept of a metric space. - a set where closeness can be measured using non-negative real numbers for the measurements.
• In Linear algebra, an inner product such as one given for poynomials where  `<f,g>  = int_a^b f(x) g(x) dx` [ integration of the product over an interval] gives rise to a norm, `||f|| = sqrt{<f,f>}`, and then a metric, `m(f,g) = ||f-g||`.

9-4
• Discussion of the mean value theorem, its proofs, and counterexamples related to its hypotheses;

### (i) continuous on [a, b] and (ii) differentiable on (a, b), (iii) f (a) = f (b) then there exists a number c in (a, b) such that f'(c) = [ f (b) - f (a)]/[b-a].

• Comment: the proof of Rolle's Theorem broke into three cases- one of which actually found the number c = (a+b)/2. This case actually did construct the desired number - and demonstrated it was a number with the required properties using the fact that the real numbers are a field with an order relation! This was connected to the first Motivtional question on the nature of the real numbers.

### Other proofs of the MVT [ Direct and geometric] were also discussed.[ See the MVT on http://www.cut-the-knot.org]

• For Monday find some theorems that relay on the Mean Value Theorem for their proof. Suggestion- proofs about increasing/decreasing/ what if f'(x) = 0 for an interval/ The fundamental theorem of calculus. [See Interactive real Analysis]and  What is the point of the mean value theorem?
• Motivational Question I: What is a number?

9-8
algebraic numbers = {z: z is a complex number which is the root of a polynomial with integer coefficients}.
• Functions:

• A few more properties of Fields:
• Suppose F is a field.
• If  a and b are in F and ab = 0 then either a =0 or b=0.

• Proof: Case 1. If a=0 then we are done.
Case 2. If a is not 0, then a has an inverse... c where ca=1. Then b=1b= (ca)b=  c(ab) = c0 = 0.
Thus either  a =0 or b=0. IRMC   [I rest my case.]
• If  a, b and c are in F a+b = a+c implies b=c.

• Proof: Let k be the element of the field where k + a = 0 Then
b = 0 + b = (k +a)+b =k +(a+b)= k +(a+c) =(k +a)+c = 0 + c = c.
• If  a, b and c are in F with  a not equal to 0 ab = ac implies b=c.

• Proof: Similar to the last result.
• Let  n·1 stand for  1 + 1 + 1 + ... + 1 with n summands. Either (i) for all n,  n·1 is not 0 in which case we say the field has "characteristic 0" , or (ii) for some n, n·1=0. In the case n·1=0, there is a smallest n for which n·1=0, in which case we say the field has "characteristic n". For example: R, Q, and C all have characteristic 0, while Z2, Zp, where p is a prime number, and any finite field such as  F4, all have a non zero characteristic.
• If the characteristic of F is not zero, then it is a prime number.

• Proof: If n is not a prime, n = rs with 1<r,s<n. Let a = 1+1+...+1 r times and b= 1+1+...+1 s times. Then ab=n·1=0, so either a = 0 or b = 0, contradicting the fact the n was supposed to be the SMALLEST natural number for which n·1=0. IRMC.
9-15

• ## Rethinking the proof of the Intermediate value theorem.

• Motivation Question II  What does limit mean? for numbers? for function values? for functions?  for sets?
• Sequences of Numbers- Convergence
• Definitions and theorems connected to algebra and inequalities.
• Visualizing a sequence of real numbers:
• With a mapping figure.
• As a graph.
• On a number line.
• Visualizing the definition of  `{lim}/{n->oo} a_n = L`
9-18 (?)
Convergence in the Reals
Properties of convergent sequences
Monotonic sequences
Subsequences
9-22 25 AND 26 We covered definitions and proofs of many results about limits of sequences and continuity. In particular the Bolzano Weierstass Theorem, and the intermediate value theorem.

10-2 Begin with proof of the extreme value theorem.
The boundedness theorems

Discuss: Composition of continuous is continuous.

Definition of derivative.
Differentiable implies continuous.
Proof of product and chain rules.
Critical point Theorem. [this completes first proof of MVT!]
Examples of
Some horrible functions for limits and continuity, including the mathematician's sine.
10-6 and 13 Begin discussion of integration and connection to continuity.

10-13 and 16 Non-sequence based limits and continuity.
Continuity of Real functions
Images of intervals

Limits of functions
The epsilon-delta definition : Equivalence of definition with sequence definition.

Open , closed, bounded. Openness and continuity.
Open and closed sets in metric spaces (consider the  real numbers with the metric given by the absolute value of the difference).[Topology]

10-20
Recall: Definition of an open set. U is open if for any `x in U` there is a `delta > 0` where N(x,`delta`)  `sub`  U.
Prop:
(i) `O/` and `R`are open sets. (ii) If  `a in R` then `(a, oo )` and `(- oo, a)` are open sets.
Fact: {a} is not an open set.
Prop: (i) If U and V are open sets,  then `U nnn V` is an open set.
(ii) If   O is a family of open sets, then `uuu O` = { `x in R : x in U` for some  `U in  O`} is an open set.
Cor. : If `a < b` , then `(a,b)` is an open set.
The family of all open sets of R is sometimes called the topology of R.

Sidenote:
A topological space is a set X together with a family of subsets `O` that satisfies the three properties:
(i) `O/ in O` and `X in O`. (ii) If  `U , V in O` ,  then `U nnn V in O`.
(iii) If   {`U_{alpha}`} is a family of sets with each `U_{alpha} in O`, then `uuu U_{alpha}` = { `x in X : x in U_{alpha}` for some  `U_{alpha}` in the family} is `in O`.

Definition of a closed set. A set C is closed if (and only if) `R - C` is open.
Prop: (i) `O/` and `R` are closed sets. (ii) If  `a in R` then `[a, oo )` and `(- oo, a]` are open sets.
Fact: {a} is a closed set.
Prop: (i) If U and V are closed sets,  then `U uuu V` is a closed set.
(ii) If   K is a family of closed sets, then `nnn K` = { `x in R : x in C` for some  `C in  K`} is a closed set.
Cor. : If `a < b` , then `[a,b]` is a closed set.

Review definitions: Given f : R `->` R, and `U sub R`
The image of U under f , f (U) =  { y `in R` :  y = f (x) for some `x in R`}and
the preimage of U under f,  `f ^{-1}(U)` = {x ` in R` : f (x) `in U`}.
Continuity and open sets
Theorem:
Given f : R `->` R, is continuous if and only if  whenever U is an open subset of R, `f ^{-1}(U)` is also an open set.

10-23 (revised 10-24)
Connectedness , Continuity, and Topological Proof of the Intermediate Value Theorem.

Def'n: A subset I of the R is an interval: If a, b `in` I  and a < x< b, then x`in`I.
Def'n: A subset S of (a metric space) R is disconnected if there are open sets U and V where
(i) U `uuu` V `sup` S; (ii)
U `nnn` S `!=` `O/` , V `nnn` S `!=` `O/` , AND V `nnn` S `nnn`U `=` `O/`
Def'n: A subset S of (a metric space) R is connected if it is not disconnected.
Thus If S is connected with
open sets U and V where
(i) U `uuu` V `sup` S;
and (ii) U `nnn` S `!=` `O/` , V `nnn` S `!=` `O/` , then V `nnn` S `nnn`U `!=` `O/`.
Theorem: A subset of R is connected if and only if it is an interval.
Connected and disconnected sets

If  f : I `->` R is continuous and I is a connected  then f (I) is connected.
Proof: (Corrected 10-24) Suppose f (I) is not connected. Then there exist U and V in R where

(i) U `uuu` V =
f (I); and (ii)  U `nnn` f (I)`!=` `O/` , V `nnn` f (I) `!=` `O/` , AND V `nnn` f (I) `nnn`U `=` `O/` . We will show this implies that I is not a connected set.
Since f is continuous for the domain I, there exists open sets U1 and V1 so that U1 `nnn` I = `f ^{-1}(U)` and
V1 `nnn` I = `f ^{-1}(V)`.
Now
(i) U1 `uuu` V1 `sup` I; and (ii) U1 `nnn` I `!=` `O/` , V1 `nnn` I `!=` `O/` , AND U1 `nnn` I `nnn`V1 `=` `O/`. [why?]
But this contradicts the assumption that I was a connected set.

IVT.

Finite subcover property.
• Compact sets(topological definition)
• Heine Borel Theorem (simplest version): `[a,b]` is a compact subset of R.
Proof: (outline). Suppose O is a family of open sets where `uu O sup [a,b]`. Let G= {x `in` `[a,b]` where `[a,x]` is covered by a finite subset of the members of O.}.  Claim ( "easy"): G
`!=` `O/` and G is bounded by `a`. Then let `alpha` = lub of G.
Show `alpha in ` G and then `alpha = b`.

• Theorem: A closed subset of a compact set is compact.
• Corollary: If K is a closed and bounded subset of R, then K is compact.
Proof: K is bounded- so K `sub [a,b]` for some `a` and `b`. But by HB, `[a,b]` is compact, so K is a closed subset of a compact set thus K is compact.
• Theorem: If K is a compact subset of R, then K is closed and bounded.
Proof: (i) K is bounded: Consider the family of open sets, O = {B(0,n) for n = 1,2,3.... }. Certainly this family covers any subset of R, so it covers K. But a finite subcover will be bounded by the largest n of the sets in that subcover. Thus K is bounded.
(ii) K is closed.  Let U = the complement of  K = KC. We show that U is open.  Suppose p `in` U  and x `in ` K.  Then let r(x) = |x-p|/3 so r(x) > 0 and B(p,r(x)) `nnn`B(x,r(x)) = `O/`. Let O be the family of open sets = {B(x, r(x)) : x `in` K}.  Then O covers K and because K is compact there a finite number of points x1, ... , xm where
the family of open sets{B(xk, r(xk)) : k = 1,2,..., m} covers K. Then let r = min{r(xk) : k = 1,2,..., m} and we have that B(p,r) `nnn`B(xk,r(xk)) = `O/` for all k, and hence B(p,r) `nnn`K = `O/` Thus B(p,r) `sub` U and U is an open set.  EOP.

Compactness and continuity.
Theorem:
If  f : I `->` R is continuous and I is a compact  then f (I) is compact.
Proof: To show
f (I) is compact, suppose O is an open cover of f (I). Because f is continuous, for each U `in` O, there is an open set U* where U*`nnn` I = `f ^{-1}` (U).  Let O* = { U* : U `in` O }.  Note that O* is open cover of I , and since I is compact, O* has a finite subcover of I.  For these (finite) open sets we have corresponding U so that U*`nnn` I = `f ^{-1}` (U).  Thus .... these U are a finite subcover of the family O.

Topological proof of the Extreme Value Theorem.

BREAK! I have fallen behind in keeping these notes up to date.
Since the discussion of campactness we have been looking at integration.
We discussed both forms of the Fundamental Theorem of Calculus and proofs.
These proofs led us to a discussion of more basic theorems about the definite integral.
We went on an excursion into the concept of uniform continuity, proving that a continuous function on a compact set of real numbers was uniformly continuous. this allowed us to prove that any continuous function on [a,b] was integrable.
We also demonstrated the additive property and monotonicity for the definite integral.
The sum and scalar multiple properties will be done in class by students in the near future.

11-13
Today we started an investigation of functions and comparing them to numbers- in particular we introduced two distinct norms on functions: the sup norm and the L2 norm.
These will be discussed later.
We turned our attention to sequences and
series
reviewing some of the properties for convergence related primarily to numbers. We noted how geometric series could be thought of as a sequence of functions.
We reviewed and proved:
•  the divergence test,
• geometric series,
• convergence of positive series,
• the comparison test,
• the integral test,
• the harmonic series,
• p- series and
• how absolute convergence implies convergence.
11-17
The Ratio and Root Tests
and other stuff.
Rearrangements of series and conditional convergence
Power series
Series and Power Series

Taylor's theorem: approximating functions with partial sums of power series
Taylor Series (with remainders - Integral & Lagrange)

Definition of uniform convergence

Uniform limits of continuous functions are continuous

Uniform limits of Riemann integrable functions

The Weierstrauss M-test

Convergence of power series

A nowhere differentiable function

Metric spaces: definition and examples
Convergence in a metric space
Convergence in infinite dimensional spaces
Properties of uniform convergence
The Weierstrass approximation theorem

12-1
Given the integers as an ordered integral domain ("commutative ring with unity and no zero divisors") we construct the rational numbers by first considering P= {(a,b): a,b integers and b not 0} and the equivalence relation on P where (a,b) ~ (c,d) [in P] if and only if ad= bc.
We proved that as a relation on P, ~ is symmetric, reflexive, and transitive. we let [a,b] = { (r,s) in P : (a,b) ~ (r,s) } and showd this partitioned P, i.e., (i) every (a,b) in P is in [a,b], and (ii) if [a,b] amd [c,d] have some element in common, then [a,b]= [c,d]  - as sets.
The rational numbers Q as a set = {[a,b]: a,b are integers and b is not 0.}

We then define operations on Q for addition and multiplication- show these operations are "well defined" and that with them Q is a field.

Finally we characterized what the positive rational numbers are and from this defined an order relation on Q.

The result made Q  an ordered field with a 1:1 function from Z to Q defined by `a ->[a,1]` that preserves the order and algebraic structures of Z.

12-4
Defining the real numbers:
we looked at ways we describe real numbers- eventually leading to a discussion of the decimal notation and its connection to infinite series using powers of 10 which converge by comparison with geometric series.
This led to a discussion of other sequences that characterize real numbers- in particular  e as the limit of the sums from the Taylor series `sum 1/(n!)` and the limit of the powers - `(1+1/n)^n`.
To eliminate the assumptions of a limit existing for these sequences of rational numbers, we can use the theory that characterizes convergence with the Cauchy condition. This is the start of a more general definition of a real number using "cauchy sequences" of rational numbers.
We let R = { S={`a_n`} : where `a_n` is a rational number for each n and  S is a Cauchy sequence }
Now we establish an equivalence relation on R , namely S~T [T = {`b_n`} in R] if for any positive rational number `epsilon` there is a natural number M where for any k >N, `|a_k - b_k| < epsilon`.
[Thus we indicated why ~ is a reflexive, symmetric, and transitive relation.]
This leads to the partition of R into equivalence classes [S]= {T in R: S~T}.  `R` = {[S]: S is in R} can then be used to define operations and an order relation  [ inherited from Q] so that `R` is an ordered field which satisfies the least upper bound property. Thus `R` is a complete ordered field- "the real numbers."

12-8
More on the definition of the real numbers:
Review:
Axioms for the Real numbers-

Construction of the real numbers- revisited on line-
including the proof of the least upper bound property.
["Close to what we outlined last class.]