I think the way people generally arrive at this conclusion is the tacit assumption that real numbers are in one-to-one correspondence with the strings representing them, acquired through inductive reasoning.

We rarely address the distinction between numbers and numerals. I think that leads people to confuse the notation with the number.

"The myth is widespread, and knowing it from both perspectives will help in quashing it." - Most people who believe irrational things don't believe them because they have a "good" argument for it. They believe them because they feel they should be true. Some might then go and find arguments supporting the belief, but even if you strike all of them down, it wouldn't actually convince them the belief was wrong, since those arguments were never the basis of the belief in the first place.

Don't need a false proof. Just need a fundamental misunderstanding. I'd say If every $a_n <1$ then $\lim_{n\to \infty} a_n < 1$. That's as valid a proof as any other false proof. And as a false proof I think it's error is exactly the same error that leads anyone to be mistaken about it.

@fleablood I don't know about your experience, but usually when I've seen it, the error comes in claiming that the limit isn't the same as the value, as if we were trying to trick them by bringing the limit into it.

They don't really have a formal reasoning. They have an intuition on infinitessimals, and an unfounded intuition that a number's decimal expression is unique. They don't understand limits - often, they have not reached calculus. There might be a non-standard analysis where the two are different, though?

@BrianMoehring I don't know whether my experience similar or different as I don't really understand the statement. "The limit of what is different than the value of what and what does that have to do with anything. But... I'd say when I've seen it does come from a not understand what limits are or, maybe this is what you mean, that limit itself can't really represent an actual thing, and if something is an actual "thing" it should be expressible directly rather than "indirectly" by a limit. I don't know. The issue is moot.

I think the best way to help people understand this particular example is to get them to try and explain very explicitly what they mean by ${0.99999......}$. Just saying "it's just ${0.999....}$ with infinitely many zeroes" isn't precise enough. Any precise definition you come up with that makes sense will lead to you concluding it's just $1$.

A single data point (yet one that I think is not uncommon): Back in college some large-ish number of years ago, one of my friends who had trouble with this insisted that if $0.999\ldots$ had an infinite number of $9$s in it, then surely by adding $0.000\text{[an infinite number of $0$s]}1$, one would obtain $1$, and since it wasn't $0$ one was adding, they had to be different. Obviously, this line of reasoning isn't sound, but I'd guess it's not rare, either.

@Riemann'sPointyNose I've been trying a new tack: If the two numbers are different, what is their midpoint on the real line? It has to be less than one and greater than the other, but there is certainly no decimal number that satisfies that. So, if they were different, we have to change our rules that let us prove if $a,b$ are distinct real number, then $(a+b)/2$ is strictly between them, or we have to say that two different real numbers have same decimal notation.

consider the limit in $\mathbb{Z}/p\mathbb{Z}$ instead of $\mathbb{R}$, then it is not true.

In general, I'm with @ThomasAndrews here: They don't have even a false proof of $0.999\ldots \not= 1$; they consider the idea that $0.999\ldots = 1$ to be the thing that is unusual and needs proof. To them, the default is that if it looks different, it is different until proven otherwise.

@BrianTung That reminds me of a book I was thinking of today, The Phantom Tollbooth. In it, the Mathematician is asked how to find the largest number, and he tells them to "follow this line forever, then turn left."

@QuýNhân What statement? You have to at least tag people so we can guess what you are responding to.

@ThomasAndrews I mean the $\lim_{n\to\infty}a_n$

But that statement doesn't make sense because we don't apply $<$ in $\mathbb Z/p\mathbb Z.$ It isn't even an ordered field. @QuýNhân We certainly don't talk about limits there.

@BrianTung The illustrator of The Phantom Tollbooth was Jules Feiffer, who just died this weekend at 95, which is why I was thinking of the book.

A question asking how to do math wrongly is not going to get much of a positive reception. You might try the philosophy stack exchange.

@DanielV: That's a fair point. Not to mention that I think the question is somewhat predicated on a misconception: that people who think that $0.999\ldots \not= 1$ have some logical argument behind it. Frankly, I'm surprised this question has gotten as much attention as it has. But I'm not sure that closing it as a duplicate is quite right, either.

Would the philosophy stack exchange actually be able to provide such mathematics? I'm not convinced. Also, where did my other comment go, where I justify this question's existence to that Martin guy? Hidden by a bigwig?

Shaun, is that a supposed duplicate? How? I don't agree that $0.999\cdots\neq1$. I'm not interested in nonstandard analysis here. I'm trying to see how other people could trick themselves into thinking it.