@leslietownes Dood I don't get it. Suppose $a \in (0,1)$ in particular $a = 1-\epsilon$ for some arbitrarily small $\epsilon>0$ then $|a_n - a| = |1 - 1-\epsilon| < \epsilon$ I guess it's not true, unless it were $\leq$

can't I just specify it as a different variable though so I have $|1-1-\delta|<\epsilon$

i would try to handle the case a = 1/2 first in very concrete terms

it can't be 1/2 since you couldn't get anything smaller than 1/2 as the epsilon

too many symbols floating around here, makes it harder to follow how or if the definition of the limit is being used, or just being stated

I'm looking at the case where $a$ is the infimum/sup of (0,1)

real analysis is nice and all but does anybody want to talk about $\infty$-categories?

okay could you write "you couldn't get anything smaller than 1/2 as the epsilon" more concretely

if $a = 1/2$ then $|a_n - \frac{1}{2}|$ where $a_n \in (-\infty,0]\cup[1,\infty)$ means $\epsilon \geq \frac{1}{2}$ since $|0-1/2| = 1/2$ and $|1-1/2| = 1/2$

like, fill in the blanks. suppose a_n converges to 1/2. putting [blank] in for epsilon in the "epsilon-N" definition of that statement, we would learn that there is a positive integer N such that [blank] for all n >= N. taking n = [blank] we would deduce [blank], which contradicts [blank]

thats anthor way of doing it

you're circling around the right idea but quantifiers are missing or not filled in. e.g. i don't know what "epsilon >= 1/2" has to do with the definition of the limit. i know it's a symbolic statement that looks kind of different from what i'm used to seeing in the definition of the limit, but it doesn't make any contradiction clear

@leslietownes you'd have lots of luck playing Russian roulette and hitting this many blanks

I don't want to hog the chat, someone talk about $\infty$-categories. I will muddle around some more and have dinner.

there's also a big conceptual distinction between applying the definition of the limit in some case to get some result, and working out or intuiting "how large epsilon could be" or whatever while still making something true, which is in general subtler, harder, and not something that working with these definitions regularly calls you to do

good luck

now, infinity-categories. fill in the blanks. if you like infinity-categories, you are a [blank] and should [blank] at the earliest opportunity

without Thorgott it'd be just Ben monologuing anyway

i still think its easier to just find the bigger interval directly

unless we have some $\infty$-categories experts here

it may not be abundantly clear how to find the best upper and lower bound, but finding an upper and lower bound is quite easy without doing any work

@leslietownes I submit "great mathematician" and "get tenure" for the blanks

throw in some cleverl use of |a| and your done

its defintly not what they want you to do though, i am just super lazy

I'm pretty satisfied with my "proof" @leslie even tho it's more like an informal argument

but I won't post it here because it's long and I want to move on because you will point out flaws :D

but this next question is prove or provide counter example to: if every a_n is rational, then a is rational.

I said: Consider $(a_n)$ given by $a_n = \sqrt{2-\frac{1}{n}}$. but I don't know if $(2-\frac{1}{n})^{1/2}$ is rational..

I'm pretty sure this is not a true statement.

@Obliv you would be correct

square roots are pretty rarely rational

OKay I will think of a different irrational number to converge to.

I know you can write out $e,\pi$ as infinite series (right?) but those aren't sequences :|

an infinite series is in particular a sequence

Is it? I thought series were sums and sequences were just a sequence of numbers

A series is a sequence of sums of numbers

$\sum_{n = 1}^\infty a_n := \lim_{k \to \infty} \sum_{n = 1}^k a_n$

there's sometimes ambiguity in textbooks over whether a "series" is the sequence of partial sums, or its limit when it exists and some unspecified other thing when it doesn't, or both, or some other thing, but certainly every series at least has associated with it a sequence of partial sums, even if it isn't literally the same thing

obliv there's the "is the if-then statement true" question which is definitely a no, and then the maybe more fun and flexible question of your favorite counterexamples and what you need to know to prove that it is a counterexample

does that mean I have to use sequence of partial sums

if you look around a bit you can probably find a series for $\pi$ or $e$ for which the partial sums are rational. Of course, the onus is then on you to show that the series converges to what you found online, and that all its partial sums are rational :)

@Obliv have to? no

e.g. there is a really quick proof that sum 1/n! is irrational, i think it's in rudin

another way you could approach this would be to think about decimals

and the proof that sum 1/n! converges, if one were required, is also really close to the surface

I was just thinking of using typical definition of rational number and playing with the numerator/denominator such that eventually the integers on the top and bottom are no longer integers or rational numbers

that's kind of garbled though, the goal would be an example where all of the a_n's are literally rational, where you can prove the sequence is convergent, and that the limit is not rational

this sounds like reducing the problem to itself. if the numerator or denominator becomes irrational in the limit, then the numerator itself is an example of the kind of sequence you're looking for

baked into that kind of approach is, how big is the universe of numbers you can prove to be irrational

@BenSteffan yeah.. i have a habit of doing that for some reason

well all do, to some extent

it's a side effect of the fact that finding meaningful insights about a given problem is hard

@leslietownes well I know $\mathbb{Q}$ is countable but $\mathbb{I}$ is not because $\mathbb{R} = \mathbb{Q}\cup\mathbb{I}$

that's true, you could give a nonconstructive proof based on cardinality if baked into what you know about R is the fact that any element of R is a limit of rational numbers

you could, but the exercise does explicitly say "provide a counter example" :)

in a classroom environment, all of these premises tend to arrive at around the same time or nearly the same time, so it's not clear what you personally can take for granted

so maybe start there - in the world of things you know, what numbers do you know are provably not rational, and how are they defined

obliv there's also the question of, are you doing this as part of an organized course of study where you can clearly delineate between what you "already know"/"can use" and what you don't, or is this problem just bubbling up out of the void

which used to be relatively rare but now seems to be the more common way that people encounter stuff, at least on main and maybe also in this chat

It's from an intro to real analysis course at my school but I don't think there's an issue of 'what I know' because I don't know very much when it comes to real analysis to begin with.

So anything you tell me, I'm learning in real time lol. The professor is very good at explaining things though so I'm kicking myself for missing two lectures shrug

oh i see

are there no lecture notes?

I haven't even looked in the textbook, that's how good they are at lecturing, but I suppose I should take a peek now.

if you know that sqrt(2) is irrational, a_n defined by a_1 = [arbitrary positive number] and a_{n+1} = (1/2) (a_n + 2/a_n) can be fairly easily proved to be a sequence of rational numbers, and slightly less easily proved to be convergent (it is eventually monotone). if you take for granted that it's convergent, algebra with "limit laws" shows you that its limit is sqrt(2)

@BenSteffan there are

then it would perhaps be a good idea to take an "inventory" of what you've covered til now

sum 1/n! is maybe simpler to analyze from a 'bare hands' approach

@Obliv well... kind of. An infinite series is like a sequence of its partial sums, but depending on context you might think of them in various ways

staring at and flipping through lecture notes is a rather integral part of how exercises are solved

Noted.

sometimes you sum without any order you see, and often times you think about a series as the limit alone

but this "A series $\sum a_n$ is like the sequence $(\sum_{i=i_0}^N a_i)_{N\geq 1}$" is a very useful perspective to have

thank you

@obliv do you know what a recurcsive sequence is?

@Obliv are you using any book?

you might benefit from, not necessarily reading, but browsing a book about real analysis

There is no reason you can't be creative in mathematics though, alot of times stupidly obvious questions have stupidly obvious solutions.

If someone askes such a silly question as a sequence of rational numbers that converge to an irrational one, no one is stop you from defining a sequence $a_n =$ the first n terms of root 2, each number is rational and the limit is obviously root 2.

you dont always need to find the hard answer to things

no, but finding this answer seems already challenging enough in this context :)

how silly a question is depends on what tools do we have

that was the point of my example, it doesnt use any tool =p

it does, digit expansions are a tool in itself

besides I was talking about it not in the sense of having tools, but how we introduce our objects and from what perspective can we look at the problem

which will be different depending on how we construct the real numbers, and which facts we have already established

also, as for silly, this question is actually profound: it justifies why $\mathbb{R} \neq \mathbb{Q}$ in the first place

in a sense its answer is the raison d'etre for $\mathbb{R}$

modulo some rephrasing

i mean all you need for that is what is a rational or irrational number, if hes not allowed to use that shouldnt be asking a question requiring its definition to be used

people who learn such topics often don't think about such details

how about topology for example

its pretty hard to claim something with a terminating finite number of terms is irrational

closed sets have different equivalent definitions, as complements of open sets for example

someone who only begins to learn topology, sometimes you assume what definitions they are using, which may or may not be true

hes clearly doing an introductory to analysis class not topology though

that was just an example

I am saying the situation is the same

yes without context every situation is the same i guess.

you'll note that some introductory classes to analysis don't give a concrete definition of $\mathbb{R}$ at all

its not uncommon, you dont really need it

you do need to introduce at least some axioms for it

yes, you do need to say something

i think mine was every convergent sequence has a limit in R

I dimly recall it being the unique ordered field as a definition

though that was a long time ago.

which puts you quite far away from convergence questions

i think it was just as an axiom though

@BenSteffan That's how I define it for my pracalc students (not in those words...).

@BenSteffan One might want to still prove that its unique with that definition. But if we aren't even proving that it exists at all then I guess one can skip that too

@Jakobian we skipped all that, yeah

"believe that such a thing exists and is unique, and if you have doubts go to the literature" :)

well even if real numbers are established, I can provide more examples

another one is exponentiation

does a student believe that we can exponentiate things, or do we prove that we can? If so, then which of the various methods do we use to do so?

There's not just one way to do it

im really not sure who your talking to?

who am I supposed to be talking to?

no idea =\

does someone talking need to be talking to someone?

no i talk to myself all the time

right. And I'm just elaborating on the previous point I made