/sci/ - Science & Math

>>11536628
I don’t know shit about math, but that makes logical sense to me

>>11536628
why couldn't you just say that last line?

>>11536659
I’m no mathematician but I think most of that other garbage is changing the equation around to be easier to work with, and specifying the limits of the proof. But that line is really all that matters.

>>11536628
Refuted.

http://aynrandlexicon.com/lexicon/infinity.html

>>11536721
nothing there is related to actual mathematics, so go fuck yourself faggot.

>>11536721
>aynrandlexicon.com

>>11536721
So which line of the proof do you disagree with?

>>11536628
>>11536929
Proposition 1 is not well defined for a finite number of digits and the [math]...[/math] symbol is not defined at all

>>11536960
It’s a sequence of digits (numbers from 0 to 9). Do you know what a sequence is?

>>11536960
>the ... symbol is not defined at all
It's definition is literally in (1).

>>11536628
yes yes, we know the LIMIT process of 0.999... =1

>>11538116
>the LIMIT process of 0.999... =1
Literally just kys at this point

>>11536628
It's been years and /sci/ is still arguing over this question.

Two numbers are different if other numbers exist between them QED

>>11538555
all you did is prove something nobody is arguing with, kill YOURself idiot. 1 =/= 0.999...

>>11536628
what you have to realize is that you're not actually proving anything, you're just redefining what "=" means

>>11538707
let's hear your definition

>>11538707
What does “=“ mean, pray tell

>>11538555
>the LIMIT process of 0.999
Kek

>>11538704
"..." implies that a limit is being taken, sweetheart. The equality holds exactly.

>>11536960
it has definitional equality as being a sequence of partial sums as given by (1). It's well defined.

>>11538707
No, it's defining what a sequence of digits after the decimal point means, you dunce.

so are the 2 arguments here
.99999 Repeating isn’t one it is infinity close to 1
And .9999 repeating will get so close to 1, it is 1

>>11539641
I’m not a mathematician so I don’t know the lingo

>>11536628
X =0.999....

10X=9.999...
- 1X=0.999...
----------------------
9X=9.00...
X=1.0...
Qed

Why is it so hard for brainlets to understand that decimals representations of rationals are not unique.

|---------|=1
|---|= 0.333....
|---------| =0.999....

>>11538726
>>11538731
>>11539615
Just as with before/after the mean square convergence for fourier series, the equal sign doesn't any longer mean "plug a value of t and watch the series converge", rather it now means that if you compute the mean square integral for a finite sum and let the degree go to infinity then that integral will tend to zero, so it is here.

Literally look at step-fucking-1. What do you think "def" means? What do you think is the point of putting the triple bar there like that?

>>11539641
>will get so close to 1, it is 1
What do you mean by "getting close to 1"? Do you mean the limit IS 1? If so, then yes, that's correct.

>>11539698
Nah, look up the definition of a limit.
>What do you think "def" means? What do you think is the point of putting the triple bar there like that?
Are you retarded? It says the string/expression on the LHS means the RHS.

>>11536628
Limits are bullshit.
Creating a number system as a bunch of “equivalence” classes of Cauchy sequences using the so-called “limit” of the differences of the sequences is also bullshit.

>>11539766
>I don't have an argument

>still not refuted
Cranks BTFO.

>>11536628
Does not follow from (6).
It doesn't matter what epsilon is, 1/10^n can always be lower when selected after epsilon

>>11540262
see
>>11540291

>>11539766
wow how will """"mathematicians"""" recover from this?

>>11539662
10X=9.999...
- 1X=0.999...
----------------------
9X=8.9999....1

>>11536628
circular logic

>>11540291
That’s the point, dummy.

>>11540291
Low IQ post dawg

>>11540445
>>11540456
I meant to say (for idiots like you that didn't infer) that it can be lower AND not zero. The very bad proof in the OP assumes it must be zero if lower. Very stupid.

>>11541048
>I don't know what a limit is

I just think it's funny that at some point someone was like okay let's represent 1/3 as .333... and did some math and changed .999... to one at the end and hundreds of years later some high schoolers were literally too dumb to accept this. Fuck you high schoolers

>>11536628
Your proof fails at the very first step (the title)

>>11541053
read the proof, sweaty

therefore lim n->inf (1/10^n) = 0 iff 1/(10^n) < E

doesn't matter what E is, 1/10^n can always undercut it without it being less than or equal to zero.

>>11541055
0.999... is equal to 1 but your "proof" is not a proof of anything. In fact, it's very close to being circular.
For your proof to work you need to
1) Prove that 1/3 is the same as 0.333..., which requires you to define what you mean by infinite decimal expressions. But if you define them in the standard way then the proof of 0.999...=1 becomes trivial and there is no reason left to jump through the hoops that you do in your proof.
2) You still have to show that 3*0.333... is 0.999.... Sure, intuitively it looks right but that's not a proof. How do you justify multiplying each digit by 3?
Perhaps you should pick up a book on analysis so you could become mathematically literate because right now you're not better than the people in the 0.999...=/=1 crowd.

>>11539766
Go ahead and define continuity without using limits or topologies, then. The fact is, the real numbers are a natural extension to straightedge-and-compass constructions. If any circle that has points on both sides of a line must intersect that line, why shouldn't the same be true for all continuous figures?

>>11541069
>doesn't matter what E is, 1/10^n can always undercut it without it being less than or equal to zero.
yeah, that's like, you know, the definition of lim 1/10^n being zero

>>11541129
>without it being zero

no wonder people make fun of mathretards

>>11541134
>>11541134
>>without it being zero
yeah, that's why, you know, the symbol "lim" is used

>>11541138
so if it is never zero, then the proof is wrong? Glad you agree anon

>>11541048
>>11541069
>>11541134
Re-read the definition of the limit of a sequence, you colossal retard.

>>11541139
1/10^n is never zero
lim 1/10^n is zero
I don't know about you, but I'm not seeing anythign weird here

>>11541072
His proof is 100% correct, though. Just because he hasn't defined what a decimal expansion means to cater to special-needs cranks here on /sci/ doesn't mean the proof is incorrect.

>>11541069
>doesn't matter what E is, 1/10^n can always undercut it
THAT'S THE FUCKING POINT YOU STUPID IDIOT

>>11541149
yet 1/10^n never becomes zero
jokes on you retard
>>11541138
yeah but if everyone agreed with that limit definition, no one would contest 0.999...=1
the limit definition is wrong

>>11541158
>1/10^n never becomes zero
nobody said it does
>the limit definition is wrong
definition cannot be wrong. definition can only be illogical or useless. definition of limit is neither.

>>11541161
okay, well I define 0.999... does not equal 1
thanks for playing my little game RETARD

>>11541166
now this is an example of an illogical and useless definition

My God, you chucklefucks and all your GODLESS "proofs", all of which start of with the basic assumption that 0.99.. equals 1, or that infinity exists, and that some fucking how, in your fantasy Universe, an infinity can be treated like some sort of really big finite number. YOU GOD CURSED SODOMITES! Guess what Heathens, have a look at this:

1=2
1-1 = 2-1
So
0 = 1

That's essentially all you are doing and you are too pig stupid to see. "Educated" fools! Starting with a false premise and then deriving a false conclusion. My God. You will burn for your sins after the RIGHTEOUS butcher you recurring decimal HEATHENS on the battlefield. DEUS VULT!

Say it with me Brothers and Sisters.
DEUS VULT! DEUS VULT! DEUS VULT!

>>11536628
Correct.

>0.999... is not equal to 1 because I can't understand the definition of 0.9999...

Cool. What application does this have for economy?

>>11536628
Holy based anon
Infinitesimal analysis is the best
(Engineers BTFO)

>>11541189
>Infinitesimal analysis
You mean analysis?

>>11536721
>trolling outside of /b/

>>11536758
shut up, furry

>>11541190
Different name here I guess
I heard all tons of shit

>>11541185
the (...) just means the 9's extend to infinity, NOT that the limit is being calculated

>>11541205
lmao it means precisely that limit is being calculated

Depends on what you mean by 0.999... if it is meant in the standard interpretation of decimal numbers to be the infinite sum given, then obviously you are correct. However, if we instead add am infinitesimal element to the reals, we may define this decimal expansion on such a manner that 0.999... ≠ 1.

>>11541207
RETARD

RETARD

RETARD

https://en.wikipedia.org/wiki/Ellipsis#In_mathematical_notation

RETARD

>Proper mathematical definition? What for?!

> Yes, I know what I mean. No, I won't tell you what I mean.

>>11541212
https://en.wikipedia.org/wiki/0.999...#Infinite_series_and_sequences

>>11541198
The whole point of real analysis is to get rid of infinitesimals.

>>11541205
>>11541212
Anon, I'll try to put really simply for you.

The real number denoted by an unending decimal expansion is, by definition, the limit of the sequence of real numbers denoted by the prefixes of that decimal expansion.

>>11541072
>Anime post
>Being retarded

>>11541241
>anon I'll put it real simple for you, every time the mathematics shows I'm wrong...I'm going to make a special note to change that result and call it a """""""""""definition""""""""""""

Face it, if you can inductively show that the number 1 will always end with a zero in its decimal expansion AND that 0.9999.... will always end with a 9
THEY CANNOT BE THE SAME THING YOU DUMB RETARD

>>11541264
Why don't you tell us how you really feel?

>>11541264
>Face it, if you can inductively show that the number 1 will always end with a zero in its decimal expansion AND that 0.9999.... will always end with a 9
you can inductively show that any FINITE decimal expansion 0.99....9 is less than 1, yes. 0.999... is not a finite decimal expansion though.
>>11541264
>THEY CANNOT BE THE SAME THING YOU DUMB RETARD
non sequitur, because limits don't preserve strict inequalities

>>11541267
SHUT UP WEEB

0.99999... OBVIOUSLY DOES NOT EQUAL 1, IT'S AS SIMPLE AS COMPARING DIGITS

JUSES CHIRST YOU FAKE ACEDEMICS ALWAYS TRY TO BE CONTRARIAN AND "UNEXPECTED" BECAUSE YOU THINK THAT'S THE SMART THING TO DO
YOU FUCKING IDIOTS DON#T YOU SEE HOW STUPID YOU'RE BEING?

>>11541271
THERE'S NO LIMIT HERE IDIOT...LOOK UP THE DEFINITION OF THE ELLIPSIS SYMBOL

>>11541278
I did

>>11541275
I can't hear you. Perhaps you could speak a little bit louder?

>>11541275
lol you so mad
>0.99999... OBVIOUSLY DOES NOT EQUAL 1, IT'S AS SIMPLE AS COMPARING DIGITS
comparing digits only gives you non-strict inequalities, because once you start talking about infinite decimal expansions, you must necessarily pass to limits. and strict inequalities aren't preserved by this step.

>>11541280
>>11541282
DUMB IDIOTS

>OH WOOPSEE I MADE A MISTAKE BETTER CHANGE THE MEANING OF THE FUCKING ELLIPSIS SYMBOL THAT#S BEEN USED FOR HUNDREDS OF FUCKING YEARS

YOU FUCKING IDIOTS CONCEDED THE ARGUMENT AS SOON AS YOU TRIED TO CHANGE THE SYMBOL

ELLIPSIS. DOES. NOT. MEAN. LIMIT YOU RETARDS

YOU WERE WRONG ALL ALONG

WRONG WRONG

W RO NG

>>11541290
You alright?

>>11541286
YOU DON#T NECESSARILY PASS TO LIMITS DUMB PSUEDO

IT HOLDS FOR ANY NUMBER OF DIGITS THAT 0.9999.... ENDS IN A 9, NO MATTER THE NUMBER. INFINITIY DOES NOT CHANGE THAT. YOU DON'T NEED A LIMIT TO TELL YOU THAT

WHAT'S NEXT "YOU MUST NECESSARILY MAKE ONE 0.9999 BECAUSE I THINK THAT'S SMART"???? GET FUCKED

>>11541297
>YOU DON#T NECESSARILY PASS TO LIMITS DUMB PSUEDO
if you want "0.999..." to be a definite real number, you do
>IT HOLDS FOR ANY NUMBER OF DIGITS THAT 0.9999.... ENDS IN A 9, NO MATTER THE NUMBER.
true
>INFINITIY DOES NOT CHANGE THAT.
it does

> 0.999.. =/= 1 morons absolutely losing their minds over a simple definition.

>>11541303
INFINITY DOES NOT IMPLY A LIMIT
INFINITY DOES NOT CHANGE THE RULES OF A NUMBER SET
INITNITY DOES NOT MODIFY THE NUMBER SET EMPLOYED
INFINITY DOES NOT MEAN WHAT YOU THINK IT DOES

>>11541310
>INFINITY DOES NOT IMPLY A LIMIT
it absolutely does
>INFINITY DOES NOT CHANGE THE RULES OF A NUMBER SET
>INITNITY DOES NOT MODIFY THE NUMBER SET EMPLOYED
gibberish without any meaning
>INFINITY DOES NOT MEAN WHAT YOU THINK IT DOES
likewise

>>11541316
Infintesimals have limits, for the purposes of the real world. Infinities, by definition, do not.

>>11541965
>Infintesimals have limits, for the purposes of the real world.
Like what?

>Infinities, by definition, do not.
Infinite series, by definition, do.

0.9+0.09+... = lim as n-> inf of sum k=1 to n of 9/10^k

>>11536685
>>11536659
brainlets, this is exactly what would be considered a full proof, no less.

>>11541264
surely it must hurt to be this dumb

>>11541264
They're different expressions that denote the same number, just like 1+1 and 2 are different expressions that denote the same number.

You do understand that different expressions can denote the same number, right?

Right?

>>11541278
>THERE'S NO LIMIT HERE IDIOT.
>>11541290
>YOU TRIED TO CHANGE THE SYMBOL
>>11541297
>YOU DON#T NECESSARILY PASS TO LIMITS
>>11541310
>[MORE AUTISTIC SCREECHING]
Let's look at the definition of a decimal expansion, shall we?
https://en.wikipedia.org/wiki/Decimal_representation
>Both notations above are, by definition, the following LIMIT of a sequence
Oopsies. You're retarded.

>>11542655
>>11541290
OHHH WHOOOOPSEE

>>11536659
Because inevitably some schizo crank will claim that [math]\lim_{n \rightarrow 0} \frac{1}{10^n} \neq 0[/math].

>>11543020
Correction: [math\lim_{n \rightarrow \infty}[/math].

>>11536659
Because inevitably some schizo crank would claim that [math]\lim_{n \rightarrow \infty} \frac{1}{10^n} \neq 0[/math].

>>11539662
Not a proof, but good for illustrating the point to people not well versed in math.

>>11539766
You could've said, "I'm a child who yells about things being bullshit instead of trying to understand them" and conveyed the same meaning.

>>11540319
Stop trolling.

>>11541175
Take your meds, religious schizo.

>>11541275
SHUT UP MATH NERDS! CLEARLY 2/2 DOES NOT EQUAL 1! THE NUMBERS ARE ALL DIFFERENT AND THERE ISN'T EVEN A FRACTION ON ONE SIDE!
That's what you sound like. That's how stupid you are.

>>11541185
>implying it's not that very definition that's in dispute
>no, they just must not get it
There is a difference between just assuming other people are stupid like it's dark matter (because it makes your thinking make sense) and unfallaciously inferring it from evidence (evidence not being that they disagree).

The inability many people today who view themselves as being "scientific" as well as math fans (left-brain prisoners) have in comprehending that .999 isn't the same as 1, is completely connected to their inability to grasp eternity and infinity. Their minds have basically been programmed to believe eternity and infinity are impossibilites. Over the course of the last 10 or so centuries of indoctrination into an alien world-view, they've "learned" to be unable to comprehend it.

This difficulty they have with eternity/infinity shows up in many different fields, from math to astrophysics.

This mental handicap is inherited directly from the (((Abrahamic))) religions, more specifically Christinsanity for us Westernerns. In it's origin, the inability to understand infinity and eternity is 100% Judaic in thought/philosophy. In contrast, the non-Jewish man; the Pagan man, at least the /European/ Pagan man, never had any problem with infinity and eternity. Christinsanity introduced into the minds of people the idea of life and the world/universe being linear, starting from point A and ending with a point B, whereas in the Native European world-view everything is infinite, a circle.

That's why many people today can't understand that .999 repeating forever will never reach 1 -- they refuse to accept the idea of an infinite/eternal repetition. Saying "it's 1" is their method of escaping from the uncomfortable (and to them insurmountable) challenge which the concept of infinity/eternity is to thier Judaically-induced mental disease.

Here's an equivalent problem for ya'll to enjoy

if you have a polygon, say a triangle, then a polygon with 1 more side, then another with 1 more side than the last, going on and on, what do you have at the end?

Every figure in the series has the property of being straight-line figures. But the limit of the sequence is a circle, clearly not straight-line.
Do you say you finish with a circle?

>>11543136
(all polygons being regular)

why is it lim n-> inf SUM (n,k=1)
not just SUM (inf,k=1) im a brainlet

>its been 10 years of these posts now and people still unironically believe 0.999...=1. holy fuck lmao

>>11543055
>samefagging your own drivel

>>11543136
You don’t finish. There is no end.

However, you could possibly define “Cauchy sequences of polygons” much like one defines Cauchy sequences of rationals to construct the notion of reals.

>>11543270
(continued.)
In that case, there’s an equivalence class of polygon sequences that approach (by some metric, say Hausdorff distance) the unit circle.

>>11543193
1/3*3=1
1/3=.33333....
.33333*3=.99999...
Therefore
1=.99999...

Go take your Cal2 faggetry elsewhere. Close is only good in horseshoes and hand grenades.

>>11543182
the first is the definition of the second

>>11536628
That's solid. Will you publish your results on arxiv?

For those who sincerely think an undefined conceptual number can be defined, here is a simple and irrefutable prediction for you all.

YOU WILL BURN IN HELL


Hear that? No matter what or how much fallacious hocus pocus you mindlessly blather, YOU GOD CURSED RAVING SODOMITES WILL BURN IN HELL!

DEUS VULT! DEUS VULT! DEUS VULT!

>>11544129
Take your meds, religious schizo.

>>11536628
basically just $$(\forall\varepsilon>0,|x-y|<\varepsilon)\Leftrightarrow(x=y)$$

>>11539641
a limit of a sequence in a set does not have to be in the set,

take 0.9999... but remove the single number '1' from the real line. All infinite members of the sequence is in the set but the limit is not.

>>11542655
>https://en.wikipedia.org/wiki/Decimal_representation
THANKS FOR PROVING ME RIGHT IDIOT

>Recurring decimal representations
>The decimal expansion of a rational number is either finite, or endlessly repeating.

>https://en.wikipedia.org/wiki/Repeating_decimal

HOHOHOHOHO WRONG WRONG WRONG
YOU'RE FUCKING WRONG
EAT YOUR WORDS YOU DUMB SLUT

>>11544365
>a limit of a sequence in a set does not have to be in the set
It does if the set is finite, and retards only believe in finite sets. If they encounter an infinite set they are forced to walk down the main street while whipping their naked upper body with a horse whip.

>>11544383
>>11544321

>>11536628
You have to first accept (1)

There are techniques for dealing with this but its not as simple as the image implies

>>11543096
10/10 shitpost anon.

>>11544383
?

Two proofs 0.999... = 1

Via Cauchy sequences:

0.999... = (0.9, 0.99, 0.999,...) = lim as n-> inf of 1-1/10^n = 1

Via Dedekind cuts:

Assume to contradiction that x is a rational number such that

0.999... < x < 1

0.999... is greater than any finite string of 9s so for any natural number n

1-1/10^n < 0.999... < x < 1

1-x < 1/10^n

10^n < 1/(1-x)

n < log(1/(1-x))

Let n = ceiling(log(1/(1-x)))+1

ceiling(log(1/(1-x)))+1 < log(1/(1-x))

This is a contradiction, so x does not exist and 0.999... = 1.

>>11544383
How does that contradict what I said, you dumb orangutan?
Face it, you were wrong. At least have the grace to admit that, you dishonest piece of shit.

>>11536628
>using limits to prove something exists
I thought asymptotes were lines the function approached but never reached

>>11544383
>The decimal expansion of a rational number is either finite, or endlessly repeating.
Yeah, like 0.999..., which is the rational number 1.

>>11545329
Come back when you’ve passed calc 1.

>0.999=/=1 fags still exist

Why does this thread never die?

>>11536628
>by definition,
>lim
nobody refutes this.

there are two answers to zenos paradox: the arrow hits the target, and the arrow never hits the target. The truth is, if you always have a garunteed space, then no matter where you stop the target is not hit. But you folks would say that the limit as the sequence goes on, the target gets closer and closer, so the lim of 1/2^x as x goes to infinity,(Which is the equation for the distance to the target at x steps, assuming the initial distance is 1), is 0. Meaning the arrow hits the target.

Those who say 0.999... != 1 don't like the idea that the arrow could even or would ever hit the target. Indeed, that's not what 0.999... is saying. 0.999... = 1 is saying that the limiting bound of the arrow's position as it goes to infinity steps is the targets position, no more and no less.

>>11543060
>saying a non-zero, non-terminating sequence is not equal to a non-zero terminating sequence is the same as saying a terminating sequence is equal to a terminating sequence

that's what you sound like. that's how stupid you are.

>>11543736

How do you multiply an infinite number of digits?

>>11545310

all you did was prove that 0.999... precedes 1 lmao dude

>>11545359
>come back when you've been indoctrinated to believe in something religiously like me

>>11543060

when people see 0.999.. = 1 what they are seeing is

infinite = finite

which is a logical contradiction.

you are never, ever going to convince anyone who thinks logically that infinite = finite.

have fun sucking cauchy's dick for the rest of your life soiboi.

I love how all the counter arguments to 0.999... != 1 boil down to either

1. we changed the definition of equality
2. we did an infinite amount of work on our algebraic proof
3. this guy said so here
4. we used a topology with closed sets

ranked by intelligence

4. trivially true but subtle as to why
2. my personal favorite
1. trivially true but not subtle
3. not trivially true, and not subtle

>>11541217
>calling it proper and mathematical makes it proper and mathematical

>>11541233
infinitesimals are just as bad a solution as closed sets.

>>11546285
>when people see 0.999.. = 1 what they are seeing is
>infinite = finite
>which is a logical contradiction.
that's their own fucking problem because that's not what the statement is fucking saying

>>11546353

cry more frenchie

>>11541271
yeah, i'm sure if we assume it will end at one point, there's no logical rule that says that at the endpoint it should end in a 9 if at every step of the way to that point all you do is add a 9 to the end, and at the last step what you do to get to the end point is add a 9 to the end

>>11544365
also
0.999999... is a periodic decimal expansion. the information content of 0.999999... is exactly log(base 10) of (1/10).

That is EXACTLY the same amount of information as the single digit "1" - that is lon(base 10) of (1/10)

>>11544365
Finally - according to brouwer (intuitionist math) 0.999999.... does NOT equal 1.

According to Hilbert, it does.

>By definition
stopped reading right there

>>11536659

then people wouldn't know you can make a simple epsilon delta proof and not think you are as smart lol

>>11546285
Nah. 0.999... is a finite number, namely 1.

>>11546315
[math]0.999 \ldots = \sum_{n=1}^\infty \frac{1}{10^n} = 1[/math]

>>11546537
Typo: [math]\frac{9}{10^n}[/math]

>>11546476
>according to brouwer (intuitionist math) 0.999999.... does NOT equal 1
Wrong again.

>>11546537
>5. we just made it up

>>11536628

I love how /sci/ is the least autistic and most blue pilled of all the 4chan boards. seriously you guys are making alan turing roll over in his grave with your non-computational bs.

>>11536628
I refute this based on the observation that 0.999 recurring does not equal 1. Get rekt math skitzos.

>>11547615
lmao ill have a go:
>0.9999999.. = 1
we have arrived at a contradiction
QED

>>11547710
Why not 1 = 2? If numbers can equal other numbers what meaning does math have?

>>11547712
1.5 to rescue

>>11536628
Give me a constructive proof you Hilbert nigger.

Refute this refutation, OP. Ball's in your court now.

claim 0.99.. = 1
def x, y = 0.9, 0.099..
thus (x + y)^n = 1 for all n
and as a consequence: PROD{n 0 to inf} (x + y)^n = 1
using the binomial formula:
PROD{n 0 to inf} SUM{k 0 to n} nCk * x^(n-k) * y^k = 1
factor out x^n:
PROD{n 0 to inf} x^n * SUM{k 0 to n} nCk * x^(-k) * y^k = 1
(i)now by the fundamental theorem of arithmetic
PROD{n 0 to inf} x^n * PROD{N 0 to inf} SUM{k 0 to n} nCk * x^(-k) * y^k = 1
(ii)and given PROD{n 0 to inf} x^n = 0 by (i) we have 0 = 1 and found a contradiction.
doubting ii, the only solution is PROD{N 0 to inf} SUM{k 0 to n} nCk * x^(-k) * y^k = 1/0 = inf such that 0 * inf = 1, which is an absurdity.
QED

>>11536628
This is a correct proof, but the definition isn't necessary.

Also, for the sake of clearity and they definition of the limit, you should say $|\frac{1}{10^n} - 0| < \varepsilon$.

Ill eat crow as i did the full computation and was proven wrong. i present the proof for the positive as such now:

x= 0.9
y = 0.099..

now by the binomial theorem:

(1/x)^n = SUM{k 0 to n} (x/y)^k * nCk

such that x + y = 1

>>11547831
not OP here,i made a calculator based proof for you compmath guys

>>11541069
Lmao, so if you choose L>0, you can choose a sufficient n such that 1/10^n < L. What could this mean? You've done most of the thinking lmao

>>11541233
Maybe nonstandard analysis. Real analysis does not avoid the usage of infinitesimals??

>>11541264
>will always end with a 9
Oh so it's finite? Cool

>its another garbage thread with a brain damaged retard arguing with himself

>>11541290
STINKY POOP STINKY HAHA STINKY