This is the Realm of Simply Beautiful Art

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Simply Beautiful Art

00:02

@Zophikel No, the limit does not exist

Zophikel

@Simply just checking had to prove something in my elementary real analysis book

Simply Beautiful Art

okay

Zophikel

@Simply you know any useful inequalities that can be helpful for proving :) that involves <

2 hours later…

Zophikel

01:41

@Deedlit need help with something :)

lethuc92.files.wordpress.com/2014/08/…

Page 11 can't figure out what to do with the n

1 hour later…

user21820

02:59

@SimplyBeautifulArt Honestly, I didn't intend any pun! But eventually one is bound to make accidental ones after talking long enough. =)

@Zophikel @SimplyBeautifulArt: Nope!!

user21820

03:32

@SimplyBeautifulArt Something's wrong with your translation. It doesn't make sense to me.

The original is:

> ∀ϵ>0 ∃δ>0 s.t. |x−a|<ϵ => |f(x)−f(a)|<δ

This is itself already not well-formed.

"x" is meaningless.

And convention in logic is to assume that x is quantified outside, which would make it even worse.

user21820

03:53

But suppose we consider instead:

> ∀ϵ>0 ∃δ>0 ∀x ( |x−a|<ϵ => |f(x)−f(a)|<δ )

> where the quantifiers are over reals of course.

Then the difference between this and normal continuity is not that the condition and consequence of the implication is switched around!

It's that the ε,δ in the implication have been switched.

And note that if you had switched the implication around as follows:

> ∀ϵ>0 ∃δ>0 ∀x ( |f(x)−f(a)|<δ => |x−a|<ϵ )

This is not equivalent to continuity of f, nor continuity of the inverse of f!

It's a good exercise to find a counter-example for each of these two claims.

3 hours later…

user21820

06:48

@Zophikel: And by the way, it seems to me that you do not understand the meaning of the logical structure, which is preventing you from grasping the definition of limits. Can you tell me the difference between "forall x ( exists y ( P(x,y) ) )" and "exists y ( forall x ( P(x,y) ) )", where "P(x,y)" is some statement about x and y?

2 hours later…

shredalert

09:16

logs on to chat, sees epsilons and deltas everywhere

YOU ALL NEED OPEN BALLS IN YOUR LIVES!

user21820

09:28

@shredalert: Alright.

Continuity of a function f : S -> T where S,T are metric spaces is expressed as follows:

> forall x in S and open ball B in T around f(x) ( exists open ball A in S around x ( f(A) within B ) ).

@shredalert: Happy now? =D

shredalert

09:46

@user21820 That's more like it. :D

The open ball and open set definitions are my favourites.

@user21820 Good afternoon btw :)

user21820

@shredalert Same to you!

shredalert

Morning here, still.

user21820

Oh you corrected for my time zone.

I didn't realize.

Lol!

shredalert

Going to study some logic after breakfast! Excited! :D

user21820

Nice!

My finger feels very itchy like it wants to star your comment.

Just to advertize logic. =P

shredalert

09:56

haha

user21820

Itch scratched. =D

shredalert

Satisfied haha

user21820

The thing I don't understand is why we can't unstar comments in chat.

We can for normal questions on SE.

So we can use them to keep track of an interesting question. But we can't use that for this purpose in chat.

Hehehe..

shredalert

Maybe they just didn't get around to implementing it yet

user21820

Perhaps, yea.

shredalert

10:00

Next two weeks will be a busy time for me. Got two assignments due in on the 26th

user21820

Ah okay

All the best for your assignments!

shredalert

Many thanks

I had such a good time on the section about attacking arguments in my logic book. Attacking premises, and the inferring arrows in arguments. The sections on reasoning are just as fun as the section on the more mathematical/formal side.

user21820

Ah nice.

It's good for people to learn to do that.

Not just math students too!

shredalert

Indeed

I'm now beginning to think logic should be given a part in the core curriculum, even in highschool. No matter what people go on to do, it will be useful.

user21820

Yes.

I've always said that, but few people believe me.

Some think it is impossible to teach younger students logic, which I strongly disagree.

shredalert

10:07

You don't realise how helpful it is until you do it, that's the problem, I guess.

user21820

Some think it's a matter of mentality and that some people will never get it.

That I even more disagree.

It's like throwing in the towel before it gets wet....

And I don't mean teaching first-order logic. I mean teaching in such a way that promotes critical logical reasoning.

No symbols needed at all.

shredalert

I'm sure students in highschool will benefit and even understand it more than calculus

user21820

Yeap! Exactly!

And furthermore when the time comes to do rigorous mathematics, then introducing the logical symbols to capture the logical meaning would be so natural that they like it.

Rather than hate it, as is common today.

shredalert

When I was doing my A-levels I got an A on one of the most difficult calculus modules, doesn't mean I understood anything. lol

I'd rather do a course I understand and am fully able to appreciate.

user21820

Yeap that's very commendable.

How are your current courses going, by the way? Learned anything you found uniquely interesting?

shredalert

10:13

I'm enoying my module on mechanics. Parametric equations were wonderful for me.

still are

Also had stuff on conic sections and geometric transformations, basically as a prelude to more advanced algebra. Those were fun as well.

I think I prefer the more non-linear side of mathematics.

user21820

Ah I see.

I only took one course on geometry, and I was intrigued by the easy proof of the Dandelin spheres.

shredalert

I didn't cover that in my course but I studied it myself

It surprised me when I found out how old it was

user21820

Haha. Yea there's lots we can do without coordinates! =)

shredalert

Exactly why I love geometric algebra

user21820

Ah.

shredalert

10:18

The little bits of it that I know now at least

I'm also having fun with discrete mathematics

It's just hard to find the time to study everything I want to haha

user21820

Haha don't worry. You already spend much time on mathematics so it doesn't really matter what you study.

shredalert

I've definitely decided to keep doing pure and applied through to my second year as well.

Need to get breakfast out of the oven. Going to shoot off now. Have a good day all! :)

user21820

Sure see you!

Simply Beautiful Art

11:13

@shredalert Lmao

Fawad

@SimplyBeautifulArt there is reinman integration problem

Please see

Q: How to evaluate this limit? Riemann Integral

Find the coordinates of the maxima of $f(x)$ First you take a natural logarithm and simplify to get an expression and then notice a Riemann SUM. Simplify this and you will get an expression independent of "n". The expression will depend only on x. Please help me out to find this and then fin...

Simply Beautiful Art

What a mess

@Zophikel Lol, my next badge is supposed to be complex analysis

1 hour later…

Simply Beautiful Art

12:50

hehe

when someone puts "lao shuai" instead of "lao shi" on their homework

Simply Beautiful Art

13:13

@shredalert Eat the logic for breakfast

user21820

@SimplyBeautifulArt I don't get the joke. It's the same word isn't it?

Simply Beautiful Art

@user21820 Lao shi means teacher. But shuai means... "cool"

so I suppose it means something weird like cool person instead of teacher

user21820

Yes but aren't they written the same?

Simply Beautiful Art

No, shi has a line on top, but shuai doesn't

user21820

Oh oops.

I failed at Chinese anyway.

Simply Beautiful Art

13:19

lol

2 hours later…

Zophikel

14:59

@SimplyBeautifulArt and @user21820 I got the results from my gifted test back

It seems like I have trouble reassembling information without looking it up

But @SimplyBeautifulArt i'm getting a mentor to learn from so i'm happy :)

anyway how was your guy's week

2 hours later…

Simply Beautiful Art

17:28

@Zophikel My week was "PAIN"-ful

Zophikel

@SimplyBeautifulArt I got the intial results for my IQ test it seems like I have a rigouts understanding of what i'm learning but I can't rigorously communicate

Simply Beautiful Art

lol

nice?

Zophikel

@Simply it's not so nice :( I'm having trouble with communicating rigoursly

Simply Beautiful Art

yeah okay

Zophikel

@Simply how did you deal with this I can't to talk to anyone rigoursly in RL

Simply Beautiful Art

17:36

lmao

How did I deal with it? Intuition and practice mate

it takes a while

but you might be able to find help in the main chat room

Zophikel

@Simply thanks I feel like i'm getting better but when I talk I feel like it's not detailed enough

Simply Beautiful Art

Mhm

$Mathematics$ Mathematics

Associated with Math.SE; for both general discussion & math qu...

I'm not gonna be here soon

^ And you might find help there

Zophikel

@Simply how come

Simply Beautiful Art

work and stuff

Zophikel

@Simply you have a job what do you do

Simply Beautiful Art

17:39

...

top secret :-)

Zophikel

@Simply :O you a NSA codebreaker or something

Simply Beautiful Art

No...

I'm a high schooler...

like wtf that'd be awesome, but no

Zophikel

@Simply i'm just joking

Simply Beautiful Art

if you wish, you could try searching through the 100,000 messages in this chat room to find out

Zophikel

@Simply 100,000 messages :(

Simply Beautiful Art

17:45

Mate, this chat room is busy

Zophikel

@Simply I thought it would be like 50,000 messages

@Simply can't wait to get a mentor

Anyway @Simply what's happening for you also do you have discord ?

2 hours later…

Zophikel

20:17

I'm having trouble with this consider the sequence: $S_{n+1}=\frac{1}{3}(s_n+1) for \, n \geq$ $S_1 = 1$. I essentially have to prove s_n > \frac{1}{2}

I'm attempting this through induction

I've got so far $P(N) = S_{1+1} = \frac{1}{3}(1 + 1) \, S_{2} = \frac{1}{3}(2)$ $\frac{2}{3} > S_n$

After establishing my base case I attempted do this for $P(n+1)$

$P(n+1) = S_{n(n+1)} = \frac{1}{3}(S_{n+1}+1) for \, (n+1) \leq 1$

^ Is this correct

1 hour later…

shredalert

21:45

@Zophikel Is $S_1$ the only term of the sequence you are given?

@SimplyBeautifulArt logic would be a pretty low calorie breakfast haha

1 hour later…

Zophikel

22:46

@shredalert I got it figured out don't worry :)

@shredalert also that post has some huge mistake forgot to establish the base case

Simply Beautiful Art

23:44

@Zophikel Yes, I has Discord

Zophikel

@Simply what is it

Simply Beautiful Art

Simple_Art
#1200

Easier induction proof stuff

$S_{n+1}=\frac13(S_n+1)>\frac13(\frac12+1)=\frac12$, so it holds if $S_n>\frac12$

with $S_1=1$, we are done.

@shredalert I eat a lot

Zophikel

ahhh all right

@Simply I initially got it when working it out

Simply Beautiful Art

okay

Zophikel

Induction proofs with recsurive sequences can get a little confusing when establishing the base case

Simply Beautiful Art

23:47

Want to know the general term?

Er, want to know how you would find the general term?

Zophikel

@SimplyBeautifulArt yeah

Simply Beautiful Art

I start by repeated expansion and pattern recognition:

Zophikel

hmmmm goo on..

Simply Beautiful Art

$S_n=\frac13S_{n-1}+\frac13$
$=\frac19S_{n-2}+\frac13+\frac19$
$=\frac1{27}S_{n-3}+\frac13+\frac19+\frac1{27}$
$=...$

Zophikel

all right

Simply Beautiful Art

23:50

Once I feel like I can guess what $S_n$ should be, then I prove the formula works by induction

Can you guess what $S_n$ might be?

Zophikel

@Simply no

Simply Beautiful Art

Consider trying to write $S_n$ in terms of $S_{n-k}$. What do you get when $k=1$? $k=2$? $k=3$? etc.

then consider when $k=n-1$, and you'll get a formula for $S_n$.

Zophikel

@Simply ahh ok I didn't think of doing it this way

Simply Beautiful Art

Also note the geometric series

Once you work it out, find the limit of $S_n$, which should be $1/2$.

Zophikel

@Simply interesting I got $1/2$ using a purly basic way

Simply Beautiful Art

23:54

How so?

(did you check convergence?)

Zophikel

@Simply let me check my book

I just let S_n=0 and went from there

@Simply I think I made a huge error

Simply Beautiful Art

XD wtf

Okay, first of all, $S_n>\frac12$

Secondly, no

Thirdly, why?

Zophikel

One I mistyped the orginal post

so hold on let me check what I wrote in my notebook

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