@Zophikel No, the limit does not exist

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

okay

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

@Deedlit need help with something :)

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

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

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

@Zophikel @SimplyBeautifulArt: Nope!!

@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.

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.

@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?

logs on to chat, sees epsilons and deltas everywhere

YOU ALL NEED OPEN BALLS IN YOUR LIVES!

@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

@user21820 That's more like it. :D

The open ball and open set definitions are my favourites.

@user21820 Good afternoon btw :)

@shredalert Same to you!

Morning here, still.

Oh you corrected for my time zone.

I didn't realize.

Lol!

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

Nice!

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

Just to advertize logic. =P

haha

Itch scratched. =D

Satisfied haha

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..

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

Perhaps, yea.

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

Ah okay

All the best for your assignments!

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.

Ah nice.

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

Not just math students too!

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.

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.

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

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.

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

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.

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.

Yeap that's very commendable.

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

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.

Ah I see.

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

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

It surprised me when I found out how old it was

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

Exactly why I love geometric algebra

Ah.

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

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

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! :)

Sure see you!

@shredalert Lmao

@SimplyBeautifulArt there is reinman integration problem

Please see

What a mess

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

hehe

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

@shredalert Eat the logic for breakfast

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

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

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

Yes but aren't they written the same?

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

Oh oops.

I failed at Chinese anyway.

lol

@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

@Zophikel My week was "PAIN"-ful

@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

lol

nice?

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

yeah okay

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

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

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

Mhm

I'm not gonna be here soon

^ And you might find help there

@Simply how come

work and stuff

@Simply you have a job what do you do

...

top secret :-)

@Simply :O you a NSA codebreaker or something

No...

I'm a high schooler...

like wtf that'd be awesome, but no

@Simply i'm just joking

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

@Simply 100,000 messages :(

Mate, this chat room is busy

@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 ?

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

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

@SimplyBeautifulArt logic would be a pretty low calorie breakfast haha

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

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

@Zophikel Yes, I has Discord

@Simply what is it

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

ahhh all right

@Simply I initially got it when working it out

okay

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

Want to know the general term?

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

@SimplyBeautifulArt yeah

I start by repeated expansion and pattern recognition:

hmmmm goo on..

all right

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?

@Simply no

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 $S_n$.

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

Also note the geometric series

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

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

How so?

(did you check convergence?)

@Simply let me check my book

I just let S_n=0 and went from there

@Simply I think I made a huge error

XD wtf

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

Secondly, no

Thirdly, why?

One I mistyped the orginal post

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