MathAnalysis1 13 Draft

8/13/2019 MathAnalysis1 13 Draft

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P r e l i

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ARE211, Fall2013

ANALYSIS1: THU, AUG 29, 2013 PRINTED: AUGUST 18, 2013 (LEC# 1)

Contents

1. Analysis 1

1.1. References 2

1.2. Countable vs Uncountable innity 2

1.3. Sequences 3

1.4. Distance/Metrics 5

1. Analysis

Heavy emphasis on proofs in this section. Many students think that the proofs are the hardest

part of Econ 201. Only way to master the art of proofs is to do a lot of them. The best topic

in which to learn how to do proofs is analysis. A secondary goal in this topic is to get you to

be comfortable jumping between different notations. When you read more formal journal articles,

every author has his/her own notation system; need to learn how to jump back and forth between

different notational systems.

Youve just spent a few weeks in Math Camp on analysis. Why do more of it? Consensus is that

a few weeks is too short a time to master the topic adequately: people talk about the re-hose

approach to teaching math; too much too quick; one can get a mechanical understanding of what

the material means, but its very hard in this short a time to develop intuitions for the concepts.1


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2 ANALYSIS1: THU, AUG 29, 2013 PRINTED: AUGUST 18, 2013 (LEC# 1)

The goal of the few lectures Ill spend on analysis is to consolidate what youve learned, help you

understand the material more intuitively.

Having said that, students in different years have different responses to math camp. Some classes

nd it excruciating and totally inadequate preparation for graduate school; others nd that it gives

them all that they need. I dont know a priori which category this classes fall into, so Ill want

feedback on this after a couple of lectures.

1.1. References

Chapter 12 in Simon-Blume

Chapter 1-2 in De La Fuente

Appendix F in MasCollel-Whinston-Green

Chapter 1 and 2: Elementary Classical Analysis, by J. Marsden

1.2. Countable vs Uncountable innity

This is a distinction thats fundamental in math but tricky to grasp at rst. Examples are easy to

understand, but its a very difficult distinction formally. Its not much more than 100 years since it

was formally proved that there really is a difference between them. Three part distinction between

sets:

(1) nite sets:

Example: {1, 2,...N }.

(2) countably innite sets.

Example: the natural numbers , denoted N , are 1,2,3,4 ..., going on for ever.

(3) uncountably innite sets.

Example: the closed unit interval , denoted [0 , 1].


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ARE211, Fall2013 3

Distinction between nite and innite sets:

Denition: A set is innite if it can be placed in 1-1 correspondence with a strict subset of itself.

Example:

{1, 2} is a strict subset of {1, 2, 3}. You cant map each element of the rst set to each

element of the second.

The even natural numbers, {2, 4, 6,...} are a strict subset of the natural numbers N . You

can map each element of the rst set to each element of the second.

Distinction between countably and uncountably innite sets:

Informal Denition: A set is countably innite if you can count its elements, i.e., you can identify

the rst element, the second element, etc

Example: Its easy to count the natural numbers, but you cant count the unit interval; for the

closed unit interval, theres a rst element, but there isnt a second one.

1.3. Sequences

A sequence is a mapping from the natural numbers to a set S , i.e., f : N S ; f (n) is the nth

element of the sequence. Typically, we suppress the functional notation: instead of writing the

image of n under f as f (n) we denote it by xn and write the sequence as {x1, x2 ,...,x n ,...}, i.e.,

f (n) = xn .

A collection {y1, y2,...,y n ,...} is a subsequence of another sequence {x1, x2,...,x n ,...} if there exists

a strictly increasing mapping : N N such that for all n N , yn = x (n ) . Note that maps

the domain of the subsequence into the domain of the original sequence. For example, consider

the sequence {3, 6,..., 3n,... } and the subsequence {6, 12,..., 6n,... }. In this case, the function we

need is (n) = 2 n, i.e., for all n, yn = x2n : e.g., y1 = x2 = 6, y2 = x4 = 12. That is, you construct a

subsequence by discarding some elements of the original sequence, but keeping an innite number

of the original elements and preserving their order.


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Its useful to compare the following kinds of mappings. The only distinction between them is the

domain of the mapping. Language Im going to use here is conventional but by no means universal.

(1) v : {1,...,N } {1}, i.e., (1,..., 1). This is more commonly refered to as an N -vector of

ones, Distinguishing feature of a vector is that the domain of the mapping is nite .

(2) x : N {1}. This is a sequence of ones, i.e., xn = 1, for all n. Distinguishing feature of a

sequence is that the domain of the mapping is countably innite .

(3) f : R + {1}. This is a continuous function , mapping the non-negative real numbers to

1, i.e., f () = 1. Distinguishing feature of a function is that the domain of the mapping is

uncountably innite .

Until youre taught to think otherwise, youd probably think of only the latter as a real function.

Actually, all three mappings satisfy the true denition of a function, i.e., each of them assigns a

unique point in the codomain to each point in its respective domain.

Some examples of sequences:

(1) {1, 2, 3, 4...}

(2) {1, 1/ 2, 4, 1/ 8... }

(3) { 1, 1, 1, 1...}

(4) {1, 1/ 2, 1/ 3, 1/ 4... }

(5) sequences arent necessarily maps from N into scalars. We could have a map from N into

the set of continuous functions. For example, consider the sequence of continuous functions

{f 1, f 2,...f n ...}, where f n =

1 if x 1/n

nx if 1/n < x < 1/n

1 if x 1/n(6) sequences dont necessarily have a closed form representation , i.e., you cant necessarily

write down a formula that expresses the sequence. For example, if you started generating

random numbers and continued forever, you would have a sequence.


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ARE211, Fall2013 5

Note that the difference between a sequence and a set of comparable size is that the order of the

elements in a sequence matters, while the order of the elements of a set does not. Thus, the

sequences {1, 2, 3, 4...} and {2, 1, 3, 4...} are different, while the sets {1, 2, 3, 4...} and {2, 1, 3, 4...}

are the same. Moreover, the set {1, 1, 1, 1...} is just the singleton set {1}, while the sequence

{1, 1, 1, 1...} is quite different from the scalar 1.

1.4. Distance/Metrics

Analysis is all about how close things are to each other. Does a sequence converge to a point?

There are lots of notions of closeness in mathematics, some of them more intuitive than others. So

long as we are considering closeness in the context of Euclidean space, most notions of closenessare essentially equivalent. However, once we get to consider closeness in the context of functions ,

there is a vast variety of quite different notions. Mathematicians have an abstract notion of what

is a legitimate measure of closeness.

Denition: a metric or distance function on a set S is a function d : S S R satisfying, for all

x, y S :

(1) d(x, y) = d(y, x ) (symmetry )

(2) d(x, y) 0 (nonnegativity )

(3) d(x, y) = 0 iff x = y (two elements are a positive distance apart iff they are different from

each other)

(4) d(x, y) d(x, z ) + d(z, y), for all z S (the triangle inequality )

The last property of a metric is the one that has the most bite, and the one that really captures

the spirit of distance: it states that the shortest distance between two points is a straight line.

Examples of metrics

(1) on R : d1(x, y) = |x y| .

(2) on R n : d2(x, y) = ni =1 (x i yi )2


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(3) on R n : d (x, y) = max {|x i yi | : i = 1 ,...,n }.

(4) on R n : d(x , y ) =1 if x = y

0 if x = y(well call this the discrete metric).

An example of a function that is not a metric is e(x , y ) = min {|x i yi | : i = 1 ,...,n }.

Lets check that the function d(x , y ) =1 if x = y

0 if x = yis indeed a metric. It clearly satises the rst

three properties. What about the triangle inequality. First observe that if x = y , then d(x , y) = 0.

Since d(x , z) + d(z, y) is necessarily nonnegative, then the triangle inequality holds. Now suppose

that x = y so that d(x , y ) = 1. In this case, for all z, either z = x or z = y in which case either

d(x, z) or d(z, y ) is 1 so the triangle inequality is again satised.

Now lets check that the function e(x, y) = min {|x i yi | : i = 1 ,...,n } is not a metric. Well it

fails the third condition, since e((1, 1), (1, 2)) = 0, but (1 , 1) = (1 , 2). More importantly, e also fails

the last condition: set x = (1 , 1), y = (2 , 2), z = (1 , 2), e((1, 1), (2, 2)) = 1 but e((1, 1), (1, 2)) =

e((1, 2), (2, 2)) = 0 so that d(x, y) > d (x , z) + d(z, y ).

When S is a space of functions , condition (3) in the above denition of a metric is too restrictive.

In economics, for example, we often encounter functions that arent equal to each other, but are

said to be of distance zero from each other. In particular, it is often natural to say that the distance

between two functions is the integral of the absolute value of the difference between them. But if

two functions differ at only a nite (indeed countable) number of points, then in this sense, the

difference between them will be zero.

Condition (3) is inconsistent with this usage. To deal with this problem, we dene a function

to be a pseudo-metric if it satises all of the conditions above except condition (3). E.g., if S

is the set of integrable functions mapping R to R , then the function : S S R dened by

(f, g ) = | fdx gdx| is a pseudo-metric but not a metric. (Note that this distance notion isquite different from the one mentioned in the preceding paragraph!)


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ARE211, Fall2013 7

(1) To see that is not a metric, consider the function f 1 dened above on p. 4 as example (5).

Because the function is so symmetric, clearly f 1dx = f 1dx = 0, so that (f 1 , f 1) = 0,but these functions are not equal to each other.

(2) On the other hand, to see that is a pseudo-metric, observe that its obviously symmetric

and non-negative. The only thing remaining to check is that it satises the triangle inequal-

ity. To prove this, we could use the following Lemma, but wont go thru it in class

Lemma: for any x, y R , |x | + |y| | x + y|.

Proof of the Lemma: Its obvious that if x and y both have the same sign then

|x | + |y| = |x + y|. Now suppose without loss of generality (w.l.o.g.) that x 0 > y .

In this case,

|x | + |y| = x + ( y) > x > |x ( y)| = |x + y|

We can now check that satises the triangle inequality. For any functions f ,g, h S ,

(f, h ) + (h, g) = fdx hdx + hdx gdxwhich from the lemma is

fdx

hdx +

hdx

gdx

= fdx gdx= (f, g )

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MathAnalysis1 13 Draft