Rudin is trivial

[hollow swallow]

So the supremum of one of them exists

For the other one (if empty) just take the “sup” as 0

[lucid swift]

I'm assuming that with K you mean like x in R such that x ≤ Re(z), by the way

[hollow swallow]

Yeah

[lucid swift]

For the annoyed Coffey that cannot infere that stuff

[hollow swallow]

Lmfao

So I thought

supA = supK + i supL

[lucid swift]

Ye, but that's isn't true

[hollow swallow]

if either of K or L are empty, replace sup with 0

[lucid swift]

Or is it

Hmm

[hollow swallow]

It is

Lmfao

[lucid swift]

They can't be empty

[hollow swallow]

(Just in case A is a subset of R)

[lucid swift]

Real numbers don't have a lower bound

How tf would it be empty

[hollow swallow]

Ok idk

[lucid swift]

Are you taking the set ${x \in \bR: x \leq -\infty}$

[hollow swallow]

We have $A\subset\bC$ right?

[wanton bronze]

Miguel

Ravi Ravioli #NoLifer

[lucid swift]

Otherwise, they're never empty

[lucid swift]

Ye

[hollow swallow]

Well then $A\subset\bR$ is also possible

[wanton bronze]

Ravi Ravioli #NoLifer

[hollow swallow]

Right?

[lucid swift]

And?

[hollow swallow]

BRUH

WHY WAS I TREATING 0 LIKE NOTHING

LMFAO

nvm me

[lucid swift]

That'd create K = {x \in \bR: x \leq 0}

Ravi learning negative numbers exist 🔥🔥

[hollow swallow]

Well not exactly that

But we’d get 0 in L

So not empty

Anyways

supK and supL exist

[lucid swift]

Yrs

Since it's never empty

[hollow swallow]

We’ll prove that supA = supK + isupL

[lucid swift]

There's always a smaller number given one

Take that number -1 for instance

[lucid swift]

Yes

[hollow swallow]

From the definition of K and L

And our order

[lucid swift]

It's easy to see it's an upper bound

[hollow swallow]

Yeah but we have a Coffey

Take an a in A

Now Re(a) <= supK from definition

[lucid swift]

Since k ≥ R(z) forall z in A, if they're equal then l ≥ Im(z) forall z in A

So it's bigger or equal than all z in A

[hollow swallow]

And Im(a) <= supL from definition

Lmao

[lucid swift]

Ye

[hollow swallow]

That was easy

[lucid swift]

Now least upper bound

Suppose there exists one smaller?

[hollow swallow]

Ah…

[hollow swallow]

Yeah

NO

fuck

[lucid swift]

;_;

Why

[hollow swallow]

The ordering :(

[lucid swift]

What happens

[hollow swallow]

Wait nvm

Wait

supK is not necessarily in K right?

[lucid swift]

ye

[hollow swallow]

That messes things up a little

But simple case division will do

[lucid swift]

Well, if it's inside it's obvious

You always deal with the other case

[hollow swallow]

Suppose there exists an a in A such that Re(a) <=

[hollow swallow]

Yeah

But if supK is outside then just supK works

[lucid swift]

Suppose there exists x + yi such that x + yi < k + li and x+yi is an upper limit

[hollow swallow]

Alr

Aka either x<y

*x<k

Or x=k, y<l

But from definition k = supK

[lucid swift]

They can be equal

[hollow swallow]

Yeah

[lucid swift]

The contradiction is in y

[crisp valley]

oh lawd

[hollow swallow]

Hmmm

Ok y<l

But l is supL

l is least-upper

but y is also least-upper

This violates uniqueness of sup

@lucid swift

There is only one supremum of a set bs

Q10 is ass

Simple algebra

Exception is 0

Q11 now

[lucid swift]

Good

Ok let's do q10

Hmm

I'll do that tomorrow

I've been yawning for hours

[hollow swallow]

This is simple algebra bro

q11 is also pretty stupid

just use the fact that for 2 non-zero real numbers a,b there exists a c such that ab = c

(Take z = a+bi = a+kai, b=ak, assume both a,b non zero)

Then prove for the other cases separately

Left as an exercise for Coffey

Ok q12

Cauchy-Schwarz spam

q13

This one’s a bit interesting

Pretty sure it boils down to a+b >= sqrt(a^2 + b^2)

[lucid swift]

I'm just too tired

Plus I don't think we even have any results of exponentiation in complex

I'm gonna assume they don't want us to prove that it is well defined for all complex numbers

[terse leaf]

Walter Rudin?

[lucid swift]

No idea what's his first name

[crisp valley]

bruh

[lucid swift]

Is it relevant?

[crisp valley]

yes

[lucid swift]

AOIHs

Why is this so boring

Do I have to multipply or what

[hollow swallow]

Let’s bomb the penguins

[hollow swallow]

Ok I managed to prove that the algebraic numbers are countable (somehow, proof is prolly sketchy as hell, will post it later)

Now ex 2.3 logically follows

As R us uncountable

And algebraic numbers are countable

And a countable set can’t be equal to an uncountable set

Basically a contradiction exercise left for the reader

(Not me owo)

[hollow swallow]

Algebraic numbers are countable

[crisp valley]

he didn't fix it.

here's a bijection

Roots of $\Sigma_{i=0}^na_ix^i\mapsto\qty(a_0,...,a_n,1),\qty(a_0,...,a_n,2),...,\qty(a_0,...,a_n,n)$

[wanton bronze]

Coffey

[crisp valley]

you can take some order on the complex numbers to assign the last numbers more precisely

for example

$a+bi<_*c+di\iff \qty(a<c)\lor\qty(a=c\land b<d)$

[wanton bronze]

Coffey

[hollow swallow]

isn't the function i defined also a
wait
actually i give up
ax+b and 2ax+2b have the same roots
Coffey how to fix

[crisp valley]

$P(x)$ and $cP(x)$ have same roots

[wanton bronze]

Coffey

[crisp valley]

what's $c? $

some rational or integer whatever

aren't those countable?

[hollow swallow]

some integer yep

yeah those are countable

wait so can we just do {P(x), 2P(x), ..., } -> P(x)

[crisp valley]

you can not

[crisp valley]

make another row coordinate for the coefficient lol

[hollow swallow]

ohhh

but then it wouldn't be a bijection from the polynomials to the roots right?
unless we tag the same coefficient at the end

(of the tuple consisting of the roots)

[crisp valley]

ehh just stick the coefficient at the very front them

[hollow swallow]

huh alr

[crisp valley]

wait no

why I that even a problem by bijection alr takes care of it

or did you make something else

[hollow swallow]

Well we could make a bijection that maps $\qty{(a_0,…,a_n,1),(a_0,…,a_n,2)}$ to $(a_0,…,a_n)$

[wanton bronze]

Ravi (No Lifer)

[hollow swallow]

Basically it relates a set of polynomials with the same roots to a single reduced polynomial, or

$\gcd(|a_0|,…,|a_n|; a_k\neq0)= 1$

In the output

[wanton bronze]

Ravi (No Lifer)

[hollow swallow]

My notation is fucked
(gcd does not work nice with 0)

[wide nest]

They have taken consturction of rationals for granted I think

[lucid swift]

Nah in the end it was okay-ish justified

But it could have been explained much better

[verbal hemlock]

I don't think they are defining rational number there they are just using it as an example

And you can define rational numbers without an order as the field of fractions of the integers

[wide nest]

They didn't define rationals in the end did they?

[wide nest]

@crisp valley

fuckin hell

file too big

it says

i had made a pdf

but i guess ill have to send it page by page

@crisp valley

wasn't able to send the single pdf so i split it in 3

So i guess i wasn't able to solve 13th and 18th

[crisp valley]

damn

k

[hollow swallow]

Damn

[hollow swallow]

Bro the proof for Q2

Uhhhh

[wide nest]

what about it

oh wait it should Q instead of Z

[hollow swallow]

How is that a proof

[wide nest]

but the rest of the proof remains the same

[hollow swallow]

Bro you have to show that all the finite polynomials are countable

Sure it’s directly implicated but

“PROVE IT” - Coffey, 2024

[wide nest]

all polynomials finite or infinite are f:N-->Q by definition right? and by definition of a function. f is a subset of NxQ. and NxQ is countable. So number of polynomial is countable and so number of finite polynomials is countable and hence algebraic numbers are countable.

[hollow swallow]

What

[wide nest]

what part is unclear

[hollow swallow]

How does a polynomial map from N to Q

[wide nest]

A polynomial with rational coefficients

[hollow swallow]

Bro you can shove a complex number in there

[wide nest]

whaa...

[hollow swallow]

Every polynomial maps from C to C by the definition

[hollow swallow]

In the question only the coefficients are integers

[wide nest]

yes

[hollow swallow]

But the value of z can be complex

[wide nest]

give me a polynomial

thats fine

[hollow swallow]

ax^2 + bx + c

[wide nest]

this polynomial corresponds to the map. f(1)=c, f(2)=b, f(3)=a

[hollow swallow]

You meant it like that-

Bruhhhhh

Lmfao

[wide nest]

lol

[hollow swallow]

Oh then union of countable sets

Ah makes sense

[wide nest]

haha i should've specified that

[hollow swallow]

Yeah you should

You don’t want Coffey breathing up your neck

[wide nest]

lmao

[hollow swallow]

I’m doing chapter 2 rn

[crisp valley]

In 2, If X is countable, it's power set need not be countable always.

3 is not a proof

5.b a limit point's neighborhood can very well not contain any limit points

take 0 as a limit point of {1/n : n in N} for example

here's an alternative proof sketch :

if a_n be a sequence of limit points

go each term we can find some b_n in E such that |a_n - b_n| < 1/n as it's a limit point

now this tells us that a_n and b_n converge towards the same limit

thus a limit point(limit) of a_n is also a limit point of b_n

so it must be in the set of limit points itself

Feel free to translate this into a more topological non-sequence proof its easy

ill read the rest after I get home

but for the closure union thing

show that

$\bigcup \clx A_i\subseteq\clx\qty(\bigcup A_i)\subseteq \clx B$

[wanton bronze]

Coffey

[crisp valley]

the last part due to the closure being the smallest closed superset of a set

[wide nest]

How.

If i am not wrong we had to show that there exist non-algebraic numbers

So if there weren't any non-algebraic real numbers then all real numbers would algebraic

that means real numbers have the same cardinality as algebraic numbers

which false because real numbers are uncountable where as algebraic numbers are countable

so there has to be a non-algebraic number

[wide nest]

Huh you mean 7. b?

[crisp valley]

ahhh right

my bad my bad

[crisp valley]

was it?

its hard to read your stuff

• its tilted

[wide nest]

yup my bad.

[wide nest]

can't you like rotate it?

[wide nest]

huh. i don't think i assumed that anywhere

[wide nest]

idk what is the question statement

[wide nest]

When the OP finally finished the book, before closing the thread the last message should be "and Rudin, infact, was not trivial"

[crisp valley]

right

imma read your proofs again

@wide nest that's not all of them

[lucid swift]

So far I haven't seen anything non trivial

I'm also stuck on chapter 1 for 3 weeks tho because too lazy to do all exercises

[hollow swallow]

That’s why i told you to skip chapter 1

Bro they ask you to fail in constructing R for the last exercise

Join me in being stuck on chapter 2

[lucid swift]

Lmao

Okay I'll skip the exercises

[hollow swallow]

JOIN ME ON BEING STUCK ON CHAPTER 2

(ok i’m on like Q9 but I’m pretty sure i did a lot of fake solve)

[lucid swift]

Lmao

[hollow swallow]

I’m guessing that i fakesolved Q7 and Q8

[wide nest]

yeah

i solved this 25

and wasn't able to solve 13 and 18

so i didn't solve 13,18,26,27,28,29,30.

[crisp valley]

what was 13 and 18

[wide nest]

18 was to consturct a perfect set with no rationals

and 13 was to create a compact set with countable limit points

[crisp valley]

ah this

what have you tried?

what about

[wide nest]

S={$\frac{1}{n}+\frac{1}{m}$ | $n,m \in \mathbb{N}}

[wanton bronze]

Pertifus | #NoLifer
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[wide nest]

Along with 0

[crisp valley]

thats not compact

[wide nest]

and 1/n

[crisp valley]

yeah thats compact

[wide nest]

So K={x| x=$\frac{1}{n}$ or 0}

[crisp valley]

?

[wide nest]

So like SUK

[wanton bronze]

Pertifus | #NoLifer

[crisp valley]

heh

yeah

so it works

[wide nest]

but how prove

[crisp valley]

prove its closed

[wide nest]

thats where i am stuck

how to prove that S doesn't have any other limit points than those in K

[crisp valley]

whatd you try

anyways heres another example

instead of 1/n use 2^-n

easier to prove

[wide nest]

oh

lmao

there was one problem

where i used 2^-n and things got complicated. you gave a small example of 1/n

and now i used 1/n and things got complicated

[crisp valley]

why cant you show those are actually the only points though

[hollow swallow]

K only has 1 limit point though

[wide nest]

No like set of limit points of SUK is preciesly K

[hollow swallow]

I haven't reached 13 yet uhhhh

I'm gonna be doing Q10 when graph theory exhausts me

[wide nest]

damn

man i love topology.

2nd chapter was OP

[hollow swallow]

yeah Seven started a study group for graph theory

except he's kinda skipping the exercises

I mean, I can't blame him

[wide nest]

oh lol

[hollow swallow]

the exercises

[wide nest]

oly. graph theory or graph theory in general?

[hollow swallow]

are literally combining Graph theory, Group theory and Linear Algebra

[hollow swallow]

in general

I think

[wide nest]

prolly

[hollow swallow]

like formalizing the solutions is absolute hell

I had to legit deploy the taxicab norm for one

[wide nest]

i hope group theory and linear algebra aren't there in olympiads haha

[hollow swallow]

then legit proved something was a normal subgroup of a permutation group with a heck ton of bijections

[wide nest]

shiz

[hollow swallow]

lmfao

[wide nest]

that is all gibberish to me with my current level of knowledge

[hollow swallow]

lmfao

oh yeah the most recent question-
I have the logic but formalizing it

[wide nest]

[hollow swallow]

well it's like a matrix A is equal to its transpose

and all it's values either 1 or 0

and the elements of the form a_(ij (i=j)) are always 0

then if we switch like rows a and b and columns a and b after that, it should preserve the eigenvectors and eigenvalues

now such a "transformation" is an isomorphism of the associated graph

[wide nest]

again

thats all gibberish to me

[hollow swallow]

lmfao

[wide nest]

i don't know even the L of linear algebra

i don't what matrix is

*know

[hollow swallow]

all ik is 3b1b's series on it

it's good for intuition

[wide nest]

Ahh that

i wathed few videos and then i left

[hollow swallow]

lmfao

I watched all of the vids

[wide nest]

My math knowledge as of know goes from
geometry, basic number theory and combinatorics, basic algebra (like polynomials nd stuff no the assbract one) then a little bit of calculus (from a HC verma and yt videos) and then directly basic topology. 💀

[hollow swallow]

lmfao

[wide nest]

yup

a leap of faith

from school maths to undergrad

[hollow swallow]

my knowledge is
geometry
basic nt, combinatorics, linear alg, group theory
calculus

[wide nest]

haha

you forgot anal

[crisp valley]

actually they are used heavily

why is ravi yapping none of the ones they did till now use more than maybe 10th grade math

for example Burnside's lemma

It's useful for problems with statements like "Two things are counted once if we can rotate one to get the other"

[wide nest]

what the hell

[crisp valley]

yeah

[wide nest]

do graders even accept such solutions

also

coffey

@crisp valley

which chapter should i do from zakon

cuz there is not explicity different chapter called topology

the first few chapters are a mixture of stuff I know and stuff i don't know

[crisp valley]

do it from Zakon too

[wide nest]

yeah which chapter

[crisp valley]

till differentiation

Rudin is actually a meme

just has some good exercises

[wide nest]

what

damn

so i should abondon rudin for now?

[crisp valley]

no

do zakon till differentiation

actually

do one chapter and see what's better for you

[hollow swallow]

Coffey, god of math

[crisp valley]

Coffey, God of math.

[wide nest]

Which chap should I do

Pls dont make me do basic set theory

And like construction ofbreals from field axioms

[crisp valley]

after set theory

start from after it

[wide nest]

Thats real numbers i think

Like consturction

Fine whatever ill start from second chap

Right away

I think i'd prefer sticking with a single book at a time

Maybe zakon then rudin or visa versa but noth both at the same time

[hollow swallow]

Seriously you should change your user to “Coffey, God of Analysis”

[crisp valley]

I'm eating a Pentonic pen's packet rn

[hollow swallow]

Spoken like a true God of Analysis

[golden void]

this was supposed to be miguel talking to himself like a true schizophrenic

y'all ruined it

[lucid swift]

Wait till I get back to the book

I've had to leave it aside for other stuff for now

I'll hopefully return to it middle august

[hollow swallow]

You forgot one thing
We all are schizophrenic

[fast dune]

How are you people doing with Putin?

Carry on

[hollow swallow]

Putin

[wide nest]

Putin is trivial

[crisp valley]

https://youtube.com/@wcwou?feature=shared

https://youtube.com/playlist?list=PLun8-Z_lTkC5uY7eACFPhq9EMcG195J9d&feature=shared

for you kiddies

doing rudin

i keep forgetting where #geonosidan-prison-complex is

[white robin]

Everything sound cooler with blaccent bro

You're gonna get me all riled up! -👱🏻‍♂️
You gon make me act up fr - 👨🏿‍🦱

Quit teasing me! -👱🏻‍♂️
Stop playin wit me shawty - 👨🏿‍🦱

[wide nest]

i like this emoji so much

Does anyone else have the intense masculine urge to pet a cat at 3am?

No?

[hollow swallow]

what the-

[fast dune]

I am a catless child

[arctic dawn]

No, I'm asleep at that time

[wide nest]

me too

but we have alot of cats in my neighbourhood

[fast dune]

I like petting dogs sometimes, but then I run and wash my hands

but I think we are off topic

our topic is Putin, people

[wide nest]

ah yes

[wide nest]

We should create a rudin study group

[hollow swallow]

It got murdered

μrdered

[fast dune]

good idea

a private study group

but I don't think people can make big commitments to a whole book

still if you had 5 people you could study this book in quiet

The best you can do in this environment is advertise the fact that the book exists

I downloaded the book now that I know it exists

and I might browse thru it later this year

among 1000's of other books

and I am glad the the disord is here for swapping such items

People do not discuss the books that I have posted

and that is ok

what I expect to do here over the years is periodically bump the threads with comments and observations and wet people's interests

sort of make them curious about these books

I believe they save the materials that I have posted and might be primed to discuss these book several years form now

be patient

I might bump this thread 5 years from now if I have a chance to get to this book

however if you do a study group, you miss the opportunity to televise the book

you could have both

[lucid swift]

I have checked some

[fast dune]

see

great

that is all I hope to do

get some awareness of the book and it's importance

Is rudin a very important book?

I get the impression that it is a textbook

[lucid swift]

It isn't historically relevant

[fast dune]

I do not read textbooks

[lucid swift]

But it is a good way of learning real analysis

[fast dune]

ok

great

[lucid swift]

So yeah, a textbook

[fast dune]

well, are there any classic textbooks?

[lucid swift]

Yes, Euclid element's is a classic textbook

[fast dune]

How does Rudin compare with Courant & Robbins What is Mathematics?

[fast dune]

OK great I agree

[lucid swift]

Rudin is classic, but in a lower level

It's a somewhat famous book among analysis teachers

[fast dune]

I am glad that the book is out there and that I am now aware of it

I would not have known that without being here

[lucid swift]

Discord can be useful

[fast dune]

and I download all of the diagrams that people have posted in my discussions

it's great

I'm surprised that I like this group so much

I just hope to get a bit more math here and there by osmosis

and it is happening

[white robin]

Ngl math ain't smth you learn with "here and there" pieces of info you actually gotta sit down and dedicate urself to a textbook

And Rudin is like the most well known math books since time immemorial

[fast dune]

OK I agree. I have sat with some books and worked on them. I should post my notebooks here. Yes, you have to concentrate. But this experience is something else. I can stare at a book for a long time and wonder if I am getting it. But coming into a discussion and pitching ideas is very fruitful because stuff comes up that was actually buried in the brain and that is an awesome experience to find out what you know.