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According to my experience many students get in school a wrong mental picture of the limit as something that is realized after infinitely many steps. They think that $0.999...$ and $\sum\limits_{n=1}^\infty\frac{1}{2^n}$ somehow reach $1$ without knowing (before I tell them) that only limits are concerned but that the limit-symbol sometimes (here for 0.999...) is left out by convention.

I try to convince them that the limit $a_0$ of the sequence $(a_n)$ is simply a number that is approximated better and better and can be proved by the usual criterion $\forall \varepsilon > 0$ $\exists k$ such that $\forall n > k: |a_0- a_n|< \varepsilon$ but in cases of strictly increasing or decreasing sequences is never reached since every term of the sequence is followed by another one.

My best explanation of this notion is the Manhatten paradox: Approximate the diagonal of the unit square by rectangular stairs of equal height and width. Double the number of stairs, i.e., halve their size. Repeat this procedure again and again. Then, according to the criterion, the limit of the length of the curve approximating the diagonal is the limit of the sequence 2, 2, 2, ..., namely 2.

If however a last term could be obtained such that for every stair its lenght and height became zero, represented by a single point, then the curve would be the diagonal with total length $\sqrt{2}$ and no points side by side in vertical or horizontal direction. Since then the cardinal number of stairs would be $\lim\limits_{n\rightarrow\infty} 2^n = \aleph_0$ this would be in disagrement with with the uncountably many points of the diagonal.

Unfortunately my prime example has the disadvantage that the limit $2$ is identical with all terms of the sequence and therefore its existence as a last or final term is hard to exorcize. Therefore I am looking for a better one.

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    $\begingroup$ Correction: "but is never [necessarily] reached since every term of the sequence is followed by another one" (Actually, this whole phrase seems problematic to me.) Regarding your question, perhaps the students will get the idea after giving them examples that involve piece-wise defined functions. I used to say things such as the limit is what you guess the function will do at the point based on circumstantial evidence, whereas the value of the function at the point is what the function actually does at the point. Perhaps you can slightly modify this for limits of sequences. $\endgroup$ – Dave L Renfro Mar 6 '18 at 11:42
  • $\begingroup$ "the limit [...] of the sequence [...] is never reached" What about the sequence 0, 0, 0, 0, ...? $\endgroup$ – Joel Reyes Noche Mar 6 '18 at 14:05
  • $\begingroup$ @Joel Reyes Noche: "The limit" has two meanings: First, it is a number. This number can be the same as one or many terms of the sequence. In my above example it is 2. Second, the limit as a final step of the sequence is never reached without violating the basic law of natural numbers, namely induction. Beyond every natural number there are infinitely many. $\endgroup$ – Wilhelm Mar 6 '18 at 14:40
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    $\begingroup$ Note that every term of the sequence $2,$ $2,$ $2,\ldots$ is followed by another term, and yet the limit is also reached. (If this isn't what you intended, then "but [missing noun] is never reached" should probably be made more explicit by filling in the missing noun.) $\endgroup$ – Dave L Renfro Mar 6 '18 at 15:10
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    $\begingroup$ This comment feels like it is more about someone complaining or opining about a particular issue of rigor (debatable even). Rather than discussing how to teach math or something else related to education. $\endgroup$ – guest Mar 8 '18 at 20:22
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If your students really think that the limit is something that is realized after infinitely many steps, I think that you are doing them a disservice trying to dissuade them. Taking the limit is effectively a transfinite operation which does just that. (My students sometimes seem to get the impression that you evaluate finitely many terms then round, especially when we talk about $0.\overline{9}$.)

One reason your diagonal approximation argument does not work is that the staircase sequence that you describe does not limit to the diagonal in the appropriate function space. That is, the limit exists as a $C^0$ (continuous) function. However, you need some notion of differentiability to define the length of a curve. In any of the function spaces which include such information, your staircase sequence diverges. Since there is no limit of the curves, you can't pass to their lengths and expect to get a sensible answer.

Also, there is no reason why the number of stairs should equal the cardinality of the interval. It is entirely possible to decompose the interval into countably many intervals. It would be the number of endpoints of intervals which is countable, not the total number of points.

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    $\begingroup$ "The sequence of curves has no limit." It does or it does not, depending on the metric being used to define distance between curves. For example, using the Hausdorff hyperspace metric, the limit of the staircase curves exists and is equal to the diagonal. $\endgroup$ – Dave L Renfro Mar 6 '18 at 15:14
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    $\begingroup$ @DaveLRenfro To further your comment, the Hausdorff hyperspace metric doesn't see dimension let alone length. $\endgroup$ – Adam Mar 6 '18 at 15:36
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    $\begingroup$ @Wilhelm If you are more careful about your argument here, you will find that this is false. This is broadly similar to what happens with the Cantor set. $\endgroup$ – Adam Mar 6 '18 at 15:39
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    $\begingroup$ Indeed, almost all compact subsets of ${\mathbb R}^n$ (in the sense of Baire category, for the hyperspace) have topological dimension zero, and even Hausdorff dimension zero. And, of course, there are plenty of compact subsets of ${\mathbb R}^n$ that have both topological and Hausdorff dimension $n$ (as well as any specified value between $0$ and $n).$ $\endgroup$ – Dave L Renfro Mar 6 '18 at 15:39
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    $\begingroup$ @Wilhelm It seems that I am not going to be able to convince you that your argument is flawed. I wish you the best. $\endgroup$ – Adam Mar 7 '18 at 21:05
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I am genuinely surprised that in a SE site populated by (among others) math education researchers, nobody has yet mentioned Tall & Vinner's landmark paper:

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational studies in mathematics, 12(2), 151-169.

(Full paper here.) Tall & Vinner use the phrase concept image to refer to "the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes. It is built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures." As such, the concept image is more than a definition, and more than even a single metaphor or mental image. As they authors point out, "As the concept image develops it need not be coherent at all times. The brain does not work that way... different stimuli can activate different parts of the concept image, developing them in a way which need not make a coherent whole."

Although the main purpose of Tall & Vinner's paper is to introduce and elaborate on the notion of "concept image", they develop their ideas in the specific context of limits and continuity, which makes this research particularly relevant for the OP's question. Some tidbits from the paper:

As we shall see shortly the concept images of limit and continuity are quite likely to contain factors which conflict with the formal concept definition. Some of these are subtle and may not even be consciously noted by the individual but they can cause confusion in dealing with the formal theory... For instance, the verbal definition of a limit "$s_n \to s$" which says "we can make $s_n$ as close to $s$ as we please, provided that we take $n$ sufficiently large" induces in many individuals the notion that $s_n$ cannot be equal to $s$ (see Schwarzenberger and Tall, 1978).

To the specific question of whether $0.999\dots = 1$, the authors note that in one study

fourteen out of 36 students claimed [that] $$\lim_{n\to\infty} \left( 1 +\frac{9}{10}+ \frac{9}{100} + \cdots + \frac{9}{10^n} \right)=2$$ but $0.\bar{9}<1$.... Clearly the two questions evoked different parts of the concept image of the limiting process. In a subsequent test the same students were asked to write the following ecimals in fractions: $$.25, .05, .\bar{3}=.333\dots, .\bar{9}=.999\dots$$ Thirteen of the fourteen who previously said $0.\bar{9}<1$ now said $0.\bar{9}=1$.

and, later

One [conflict factor] of great significance is that students often form a concept image of "$s_n\to s$" to imply $s_n$ approaches $s$, but never actually reaches there... Thus [for these students] "point nine recurring" is not equal to one because the process of getting closer to one goes on for ever without one ever being completed.

Now to some extent all of the above is unhelpful if the question is "How to correct a wrong mental picture?" Nowhere in Tall & Vinner is an attempt made to fix the problems caused by conflicting concept images; their goals were to describe the phenomenon, provide language for discussing it, theorize about the origins and nature of the problem, and document evidence for it existing. But I would strongly suggest beginning there.

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    $\begingroup$ @Wilhelm In fact I think it is you who are experiencing the conflict among different concept images. In the standard reals, each finite truncation of $0.\bar{9}$ is less than 1, but $0.\bar{9}$ is itself equal to 1. As others have said repeatedly in the comments, $0.\bar{9}$ is the limit of the sequence; that is what the notation means. There is no need to take the limit of the limit. $\endgroup$ – mweiss Mar 8 '18 at 21:25
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    $\begingroup$ It is roughly analogous (mutatis mutandis) to the fact that $\frac{3}{5}$ is not merely an instruction to divide 3 by 5, but rather is the result of dividing 3 by 5. Asking for "the limit of $0.\bar{9}$" is like asking for "the answer to $\frac{3}{5}$" -- it indicates a misunderstanding of what the notation is describing. $\endgroup$ – mweiss Mar 8 '18 at 21:26
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    $\begingroup$ "Reached" is itself a meaningless term, an artifact of the metaphors of time and motion that we use to describe sequences, but not really intrinsic to the definitions themselves. (See link.springer.com/chapter/10.1007/0-387-29831-2_9). Either $0.\bar{9}$ is 1, or it is not. It's not moving or approaching or changing. $\endgroup$ – mweiss Mar 8 '18 at 21:50
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    $\begingroup$ @Wilhelm That is not the standard definition of "series"; if you have evidence to the contrary, please provide a citation to a source that supports your claim. The standard convention is that the word "series" refers not to a sequence of partial sums, but rather to the limit of such a sequence (if it exists). $\endgroup$ – mweiss Mar 8 '18 at 22:34
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    $\begingroup$ @Wilhelm There's an old saying: "If everyone in the room says you're drunk, lie down." In this case, the analogous maxim might be: "If everyone in an online forum says that you are wrong about something, enertain the possibility that perhaps it's not the other people who don't understand a subtle distinction." $\endgroup$ – mweiss Mar 8 '18 at 23:29
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I disagree with your premise. That is, I think (as in Zeno's paradoxes) that one should imagine that the limit is reached, albeit not necessarily in finitely-many steps.

Indeed, otherwise, I fail to see the interest in the notion of "limit"...

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  • $\begingroup$ In analysis the limit is a real number that satisfies a precisely defined criterion (see second paragraph of my question) or that at least can be proved to exist by the so-called Cauchy criterion. There is nothing else connected with this notion. Concerning "finitely many steps": Of course a sequence is infinite because the set of indices is not finite. There is always a next one. But the limit cannot be "reached" (i.e., realized by the sequence) because the axioms tell us the property of induction that holds for every natural number: There is always a next one, in fact even infinitely many. $\endgroup$ – Wilhelm Mar 7 '18 at 18:44
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    $\begingroup$ Formally, perhaps "there is nothing else connected with this notion", but the idea of "limit" existed non-formally for a long time prior to any formalization, and existed because it was useful, with or without formalization. When we say that a limit equals something, the intuition is that that something "is reached" (no matter what the formal definitions may be). We wouldn't care about a formalization if there were no intuition. $\endgroup$ – paul garrett Mar 7 '18 at 19:14
  • $\begingroup$ Intuition is not always a good guide. The limit is a real number that emerges from a certain definition. If your intuition says it can be reached, what does the same intuition say in case of the infinite, i.e., never ending sequence of natural numbers? There we have never a limit reached. $\endgroup$ – Wilhelm Mar 8 '18 at 11:28
  • $\begingroup$ Also, I think that much of "learning mathematics" consists of "refining one's intuition" as one encounters phenomena not well explained by one's previous intuition. $\endgroup$ – paul garrett Mar 8 '18 at 14:01
  • $\begingroup$ To add to what @paul garrett said, this image is instructive. $\endgroup$ – Dave L Renfro Mar 8 '18 at 14:12
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I don't think the issue is that they are thinking of them as where you "reach" after an infinite number of steps, as much as they need to understand that that phrasing has a lot of subtlety. Do remember that one of the major rationales for calculus is that it can use limits to solve Zeno's Paradox, and can do so in a sufficiently rigorous manner (rigorous enough to put spaceships in orbit!)

If they're thinking of limits of sequences, consider teaching them how strange infinities can behave. The Hilbert Hotel is, in my opinion, an excellent test case showing that infinities act in ways that fundamentally challenge your assumptions that come from finite numbers. Once they see how "ill behaved" naked infinities can be, they may appreciate the rigorous constraints of limit notation. It has the advantage of being a very bold series of problems. There's no subtlety here, just a set of infinities acting very much unlike you expect them to at first glance.

They may also appreciate the idea that the real numbers $1$ and $0.\overline{9}$ are actually the same number, expressed with two different sequences, but they probably need to appreciate how wonky infinities can behave before they appreciate the subtlety there. (given the typical meaning of an overbar in a decimal, you need a concept of infinitesimals to tell $1$ and $0.\overline{9}$ apart)

Edit: From the comments below, it seemed useful to post a proof that $0.\overline 9 = 1$. I have chosen this particular proof because contains nothing but simple algebra, packaging up all the interesting stuff into two assumptions.

Assumption 1: $10\cdot0.\overline 9 = 9.\overline 9$
Assumption 2: $9.\overline 9 = 9 + 0.\overline 9$

Proof:
$\text{Let }x = 0.\overline 9$
$10x = 10\cdot0.\overline 9$
$10x = 9.\overline 9$ (using assumption 1)
$10x - x = 9.\overline 9 - x$
$9x = 9.\overline 9 - x$
$9x = 9.\overline 9 - 0.\overline 9$
$9x = (9 + 0.\overline 9) - 0.\overline 9$ (using assumption 2)
$9x = 9 + (0.\overline 9 - 0.\overline 9)$
$9x = 9 + (0.\overline 9 - 0.\overline 9)$
$9x = 9 + (0)$
$9x = 9$
$x = 1$
$0.\overline 9 = 1$

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    $\begingroup$ The issue is that "0.999..." is not "a number" but a way to represent a number that hints at a way to construct it. Technically "1.0" is just a way to represent a number as well. The limit that you are referring to is actually baked into the formal definition of what 0.999... means in terms of how to construct a real number. The limit, which is 1, is actually written into the way the number is constructed. $\endgroup$ – Cort Ammon Mar 7 '18 at 19:31
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    $\begingroup$ @CortAmmon In case you haven't noticed it yet: "Wilhelm" is our dear Wolfgang Mückenheim. Google is your friend. $\endgroup$ – Uwe Mar 7 '18 at 20:10
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    $\begingroup$ @Wilhelm We have had this conversation before. The notation 0.999... does not stand for the sequence 0.9, 0.99, 0.999, 0.9999, etc. It stands for the limit of that sequence. That is the only meaning of these infinite decimals. So 0.999... does represent the same number as 1. Is it your contention that 0.999... doesn't even represent a number? It is just a sequence? $\endgroup$ – Steven Gubkin Mar 7 '18 at 20:29
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    $\begingroup$ 0.9+0.09+0.009+... is not a sequence. It is an infinite series. The value of an infinite series is the limit of the sequence of partial sums. I taught this to my freshman calculus students yesterday. This distinction between a sequence and a series is important and fairly basic. Is 7.00000... a sequence as well? It is not a sequence at all, but a number, and that number is 7. Sure, it can be viewed as the limit of the sequence of partial sums $7$, $7+\frac{0}{10}$, $7+\frac{0}{10}+\frac{0}{10^2}$, etc. But the number is not the sequence. The number is the limit of the sequence. $\endgroup$ – Steven Gubkin Mar 8 '18 at 12:38
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    $\begingroup$ No, I am not trying to answer your question in the original post. I am just glad that you now disagree with the first sentence your wrote in this comment thread "1 and 0.999... are not the same numbers since for every digit $9_i$ at position $i$ we can prove that is is not sufficient to produce 1." $\endgroup$ – Steven Gubkin Mar 8 '18 at 15:46
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Based on the comments on my first answer, I think I can take a stab at this and help out more. There are two issues here. One is the issue of the student's understanding of the need for the concept of limits, and the other is your understanding of the notation that is being used. Your interpretation of $0.\overline 9$ and $\sum\limits_{n=1}^\infty\frac{1}{2^n}$ is not the standard interpretation of those expressions, and that non-standard interpretation is causing additional confusion. Accordingly, I will avoid using those notations until late in the answer, so that we can have agreement.

The key concept that I believe your students are missing is that there is no ultimate (final, last, etc.) element to an infinite series. I believe your students have a sense that there is an ultimate element which is, as you say, "realized." Finite series have an ultimate element, but infinite ones do not.

So the challenge is to convince them of that. Rather than using complex approaches like the Manhattan paradox, stick to the simplest and most famous: Zeno's Paradox. To run a race, you must first reach halfway to your goal. From that point, you must reach halfway to the goal again, and again, and again. Zeno argued that you never reach the goal, because you always have at least one more step in this process. But obviously every person alive believes you can run somewhere, so there is clearly a problem with our thinking: our thinking claims it is impossible to do something which we know to be possible!

If we formalize this paradox, let $d$ be a series which represents how far has been traveled, such that the $n$th element of the series is how far we have traveled after we have reached $n$ halfway points:

$\{\frac{1}{2}, (\frac{1}{2}+\frac{1}{4}), (\frac{1}{2}+\frac{1}{4}+\frac{1}{8}), \ldots\}$

Now, your argument is the same as Zenos. You argue that this "process" can continue forever. If your students believe they can achieve $1$, then ask them which element of the set is equal to $1$. The goal here is to get them to phrase it as something like "the last element."

If they say something like "the infinity-th element" then you will have to work with them to understand that infinity is not a natural number. This is obviously true from set theory, but you may not want to go into set theory. You may have to approach them in an intuitive sense. If the students go down this approach, they have a concept of what infinity is, and how it behaves. This would be a key concept to teach them how to unlearn. I'd query them as to what the result of some expressions are. For example, most people I've worked with who have learned an informal concept of infinity have the idea that $2\infty =\infty$. If they have that assumption, then demonstrating the difficulty of reconciling $\frac{2\infty}{\infty}=\frac{\infty}{\infty}=1$ versus $\frac{2\infty}{\infty}=2\frac{\infty}{\infty}=2$ may be enough to demonstrate that infinity is, indeed, not acting like any number. If your students have a different concept of infinity, you as a teacher will need to learn what that concept is and provide the correction. Perhaps they will be intrigued by the puzzle of whether infinity is odd or even, given that all natural numbers are either odd or even. My expectation is that this will be an easier issue to resolve than the original issue in your question.

So once we get the students to use the phrasing "ultimate element" or something like that, then we can start to cut away at the concept that there is an ultimate element of an infinite series. I think the easiest example is the case we discussed on my other answer, $\{0.9, 0.99, 0.999, 0.999, 0.9999, \ldots\}$. As you have pointed out, this construction consists only of the digits $0$ and $9$. There is no way any element of this series can contain other digits, but if an ultimate element or last element did exist, its very easy to adapt my proof of \$0\overline 9=1$ to show that if you let "S" equal the "last element of that infinite series," my proof works. This creates a contradiction: we can prove that every element in the series consists of only 0's and 9's, but the last element (if you claim one exists) must equal 1. This contradiction will hopefully help the students abandon the idea that a last element of an infinite series exists at all.

I believe this is the intuitive challenge that your students need to overcome. They must become comfortable with the idea that there are series that have no ultimate element. I think this is the root cause of the issue you are trying to correct in your students.

Once you have that, the final step is limits. We all know that we can run places. We know Zeno's claim about the world is false, but it appears to be true about the math. How do we resolve that? The answer, of course, is what you want to teach: limits. The concept of the limit was invented because everyone had the intuitive idea that we can go places, so we should invent a concept which lines up with that intuitive understanding of reality. I'm sure many solutions were invented, but the concept of the limit proved to be the one that made the most sense and was most consistent. Eventually that was formalized with epsilon-delta proofs in 1821, but it was used in varying forms up to that point.

In Newtonian physics, there are implicit limits everywhere. We can easily take the limit of the series $d_n$, which is a finite series whose final element contains the distance traveled after reaching $n$ halfway points, as $n\to \infty$, and see that that limit equals 1. This matches up with our intuition, satisfies the mathematics, and if you look at the mathematics for an object traveling with a constant velocity, and do Riemann integration, you see that it can be visualized as a set of boxes very similar to Zeno's fractions.

So that's the selling point. The selling point is that the intuitive concepts which we try to use to resolve paradoxes like Zeno's paradox depend on either infinity being a number or on there being an ultimate element of an infinite series. But both of those assumptions bring about disastrous contradictions, so both must be false. But we all know we can run somewhere. The solution to this was the invention of the limit, which tied this mathematical problem back to reality.

Now, you will note that I explicitly wrote out every sequence in that explanation. I didn't rely on any shorthands. When you want to introduce notations like $\sum\limits_{n=1}^\infty\frac{1}{2^n}$, this is where you will need to be precise, or you will confuse the daylights out of your students.

Take the summation symbol, $\sum\limits_{n=A}^B$. In that notation, $A$ and $B$ must be natural numbers, and n basically counts from A to B. But in the glyphs above, there is an $\infty$ sign where $B$ was. Since $\infty$ is not a natural number, this is not a valid summation, by the above definition. If we wish to use that notation, we must extend its meaning. And that is exactly what mainstream mathematics has done. We define $\sum\limits_{n=A}^\infty f(n) \equiv \lim_{B\to\infty}\sum\limits_{n=A}^B f(n)$. That definition reached mainstream acceptance because it so closely matches the intuition of most people. So thusly we can say \sum\limits_{n=1}^\infty\frac{1}{2^n} = 1$ because the concept of the limit is fundamentally baked into the use of infinite summation notation. This notation has worldwide acceptance, so your students should learn it.

Likewise, the meaning of the overbar notation for repeated decimals is defined to have a limit in it. If I have the symbol $0.\overline{XYZ_n}$, where $XYZ_n$ is a finite series of n digits, then we define $0.\overline{XYZ_n} \equiv \lim_{m\to \infty}\sum\limits_{p=1}^{m}XYZ_n\frac{1}{10^{np}}$. For our example of $0.\overline 9$, this means $0.\overline 9=\lim_{m\to \infty}\sum\limits_{p=1}^{m}9\frac{1}{10^p}=1$ Again, the concept of the limit was baked into the definition of the symbology. This was done because it helped peoples intuition line up with the precise mathematics, which included a limit.

If you teach them that $0.\overline 9 \neq1$, then you put them in a bit of a bind as students. The greater mathematical community has widely accepted the above definition for what the symbols "$0.\overline 9$" mean. It is almost universal. Instead of challenging that, work with the students to teach them what that symbology means, and show that the limit appears inside the meaning of that symbology.

Just remember, your student's intuition that $\sum\limits_{n=1}^\infty\frac{1}{2^n}$ should equal $1$ is not wrong. It's not bad. That intuition is actually what drove several thousand years of mathematical efforts through arithmetic and set theory to cause us to finally reach the concept of limits. The point of limits is to satisfy that intuition.

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  • $\begingroup$ Thank you for the long answer. I would like to correct two points. 1) I do not teach that 0.999... = 1 is wrong. But I explain that in order to get the correct result, the limit has to be taken. Every mathematician knows this, but newbies in general do not. And by no means I want to make them believe that this 1 would be a final state. On the contrary. 2) Zeno's paradox has nothing to do with this problem. It is simply not applicable to divide the distance in an infinite set of halves and to expect a reasonable result. Alexander wins - but not as the result of a limit process. $\endgroup$ – Wilhelm Mar 8 '18 at 21:57
  • $\begingroup$ @Wilhelm In that case, the issue may just be what you have said here on Stack Exchange, rather than in the classroom. You made the statement that $0.999\ldots \neq 1$ on my other answer, and steadfastly defended that claim. If you are not making such claims in the classroom, then I may have drawn too much attention to it. In such a case, I'd recommend rethinking Zeno's paradox. Zeno's paradox is intimately entwined with $\{\frac{1}{2}, (\frac{1}{2}+\frac{1}{4}), (\frac{1}{2}+\frac{1}{4}+\frac{1}{8}), \ldots\}$ and $\sum\limits_{n=1}^\infty\frac{1}{2^n}$, and is a very easy to understand... $\endgroup$ – Cort Ammon Mar 8 '18 at 22:32
  • $\begingroup$ ... argument which shows that if we make naive assumptions about how that sequence "ends," we arrive at contradictions, indicating our assumptions were faulty. Thus the formal concept of "limits" is simply a mathematically rigorous formalism for our intuition (that the race can end), and what's impressive to me about limits is that they do so in a consistent way -- no contradictions form. $\endgroup$ – Cort Ammon Mar 8 '18 at 22:34
  • $\begingroup$ Once we realize that "the final element of an infinite sequence" makes no sense, we realize it's not that the final element of {0.9, 0.99, 0.999...} contains all 9's, or that the final element is 1.0, it's that the concept of "the final element of an infinite sequence" is flawed, and assuming it isn't flawed leads to contradictions. $\endgroup$ – Cort Ammon Mar 8 '18 at 22:37
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    $\begingroup$ @Wilhelm I do think that, from a math educator's perspective, it's worth reflecting on that argument. From the comments and answers here, it looks like professionals who already understand the concept of limits are confused by the way you are approaching that concept. For the sake of your students, it seems useful to reflect on that. Maybe it's just a translation thing. To me, the phrase "without applying the limit, 0.999..." is nonsensical. I would choose the phrasing "'0.999...' has no meaning that does not include taking a limit," and perhaps challenge the students to come up... $\endgroup$ – Cort Ammon Mar 9 '18 at 15:50
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The OP claims the limit symbol is left out. But the limit symbol, thus taking the limit, is not left out by convention. Its still there. Its part of the definition of $\sum_{n=1}^\infty$ which is defined as follows:

$$\sum_{k=1}^\infty a_k = \lim_{n\rightarrow\infty} \sum_{k=1}^n a_n = a$$

The infinite sum is defined as the limit of its partial sums. That the partial sums are different from $a$ is to expect when some summands $a_k$ are non zero.

The symbol $\infty$ in the infinite sum is compound to the symbol that is defined and doesn't refer to some point. The standard is that the limit operator is invoked for an infinite sum.

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    $\begingroup$ @Wilhelm your comment doesn't make any sense, since I wrote in my answer "That the partial sums are different from a is to expect when some summands ak are non zero." Also your comment is irrelevant link spam to your book and some rational wiki nonsense. $\endgroup$ – Transfinite Numbers Mar 23 '18 at 12:16
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    $\begingroup$ @j4nbur53 If you had read other texts by Wolfgang Mückenheim ("Wilhelm"), say his attempts to refute Cantor's uncountability proof for the reals and Cantor's countability proof for the rationals (sic!), the fact that his comment doesn't make any sense wouldn't surprise you. $\endgroup$ – Uwe Mar 23 '18 at 12:25
  • $\begingroup$ @Jan Burse + Uwe: "The OP claims the limit symbol is left out." I have explained that this is only the fact for 0.999... If you cannot understand that, it is your problem. But perhaps you can better understand the claims of the cranks in MathOverflow. $\endgroup$ – Wilhelm Mar 23 '18 at 15:22
  • $\begingroup$ @Wilhelm: Its also not left out for 0.999... because 0.999... means sum_k=1^oo 9/10^k. Just lookup the standard definition of sum_k=1^oo as posted in my answer. Otherwise 0.999... would be only a string and not a real number. Thats possibly a halucination of yours. $\endgroup$ – Transfinite Numbers Mar 23 '18 at 17:19
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    $\begingroup$ @Jan Burse: You say that 0.999... means the limit but that is not written and is usually not known by pupils. On the contrary, it is often claimed that all partial sums and all indices somehow exist and can be exhausted. That is contradicting the successor axiom and should be branded as unmathematical. On the other hand this idea is required by set theory. That's why this question got so many down votes. $\endgroup$ – Wilhelm Mar 23 '18 at 20:43
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It seems to me that with infinite series there are several distinct concepts and would be good for the students (and everyone) to be able to distinguish between them:

  • a finite sum of infinitely many numbers (cf. Zeno)
  • calculating what that sum must be, as an equivalent limit
  • the meaning of the notation $\sum_{n=0}^\infty x_n$
  • how to safely manipulate infinite sums (by manipulating limits instead).

My issue is that if we say the infinite sum of all the $x_n$ doesn't actually exist, only the limit, we're perilously close to saying (or maybe actually saying) that the infinite set $\{x_n:n\in \mathbb N\}$ doesn't exist either. Or for that matter, any infinite set.

I'd say the accurate picture is:

  • infinite sums exist, but not every infinite series has one.
  • they can usually only be calculated via limits. (With exceptions, such as geometric series.)
  • $\sum_{n=0}^\infty$ represents the result of taking the limit if one exists, not an instruction to "start at $0$ and stop at $\infty$"

—and "stop at $\infty$" is the self-contradictorty concept we need to avoid acquiring.

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