Does it make any sense to prove $0.999\ldots=1$?

I have read this post which contains many proofs of $0.999\ldots=1$.

Background

The main motivation of the question was philosophical and not mathematical. If you read the next section of the post then you will see that I have asked for a "meaning" of the symbol $0.999\ldots$ other than defining it to be $1$. Now here is a epistemological problem and this is mainly the problem from which the question arose. Suppose you know that $1$ is a real number. Now I give you a symbol, say $0.999\ldots$ which from now on I will denote as $x$. Now I ask you whether $x$ is a real number. To answer this, if you define $x=1$ then you are already attributing the properties of $1$ to $x$ among which one is it being a real number without proving whether to $x$ we can indeed attribute the properties of $1$.

A common response to this question has been to define the symbol $x$ as the limit of the sequence $\left(\displaystyle\sum_{i=1}^n \dfrac{9}{10^i}\right)_{n\in\mathbb{N}}$ and then prove that the limit of this sum is indeed $1$. But again the problem is that you are defining the symbol $x$ to be a real number and hence are assuming a priori that the symbol $x$ denotes a real number.

As per the discussion that has been conducted with Simply Beautiful Art let me state my position again in brief,

Also let me say that I do not disallow $0.999…$ to be a real number. My impression that if you assume $0.999…$ to be a real number then there is no sense in proving that $0.999…$ is indeed equal to $1$ because either you define it to be $1$ or you prove the equality as a theorem. But if you are going to use the limit definition of $0.999…$ then what you are a priori assuming it to be a real number which is an assumption that I don't allow. What can be allowed is that $0.999…$ is a number (but not necessarily a real number).

Question

My question is,

Does it make any sense to prove this equality?

Can one give any "meaning" of the symbol $0.999\ldots$ other than defining it to be $1$?

We have to agree about what the symbols

$$ 0.99999\dots $$

are supposed to mean. The symbols capture an intuitive idea, but it doesn't have meaning unless we agree on what that meaning is. When you write these symbols down, everyone will agree that what you mean is the following:

$$ .9 + .09 + .009 + \dots = \frac{9}{10} + \frac{9}{100} + \frac{9}{100} + \dots = \sum_{i = 1}^\infty \frac{9}{10^i} $$

There is no proof of that -- it is an agreement.

If you wrote this, perhaps similar-looking, string of symbols down

$$ \begin{aligned} 0.00000\dots1\\ \end{aligned} $$

then there is no agreement on what you mean. You would seem to be talking about a real number which is smaller than all other real numbers -- an object that doesn't exist.

So what string of symbols means what, rigorously, is a matter of agreement. Usually we elide that fact when it seems intuitive what we mean, but we do not all share the same intuition (or the same knowledge of how to express that intuition), so we must drag it out on occasions like now.

Anyway, the content of a proof of $0.9999\dots = 1$ is not that we must agree to define $0.9999\dots$ as $1$. The content is to define $0.9999\dots$ as the sum above, then by deduction show this sum is equal to $1$.

Many of the OP's questions in the comments (both to his own question and to Eli Rose's answer) keep circling back to the question "Are you assuming that $0.999\dots$ is a real number"?

The answer is no, we are not assuming it -- it can be proven. More generally, the following theorem can be proven:

Let $(a_1,a_2,a_3,\dots)$ be any sequence of numbers where each $a_i$ is chosen from the set $\{0,1,2,\dots,9\}$. Then the sequence $$0.a_1, \space 0.a_1a_2, \space 0.a_1a_2a_3,\dots$$ converges to a unique real number.

Again I want to stress that the theorem above is not assumed; it can be proven.

The notation $0.999\dots$ denotes the unique real number that is the limit of the sequence $$0.9, \space 0.99, \space 0.999, \space 0.9999\dots$$ This is just an individual instance of the general case considered in the theorem. We know that such a limit exists by the theorem , so there is no need to assume that $0.999\dots$ is a real number.

Once we know that $0.999\dots$ is a real number, and that in particular it is the limit of the sequence above, we can observe that this particular sequence converges to $1$. Since the theorem says that the limit of the sequence is unique, that proves that $0.999\dots \space = 1$.

The OP asked whether one can assign any meaning to the symbol $0.999\ldots$ other than defining it to be $1$. That question cannot be answered without analyzing what informal pre-mathematical meaning is assigned to $0.999\ldots$, prior to interpreting it in a formal mathematical sense. This of course can only be known to the OP himself but judging from the level of the OP's questions the OP seems to be a student and perhaps a freshman; see, e.g., here.

Now beginning students often informally describe this as "zero, dot, followed by infinitely many $9$s", or something similar. Such a description of course does not refer to any sophisticated number system such as the real numbers, since at this level the students will typically not have been exposed to such mathematical abstractions, involving as they do equivalence classes of Cauchy sequences, Dedekind cuts, and the like.

It is also known that at this level, about $80\%$ of the students feel that such an object necessarily falls a little bit short of $1$. The question is whether such intuitions are necessarily erroneous, or whether they could find a mathematically rigorous interpretation in the context of a suitable number system.

An article by R. Ely in this publication in a leading education journal argues that such intuitions are not necessarily mathematically erroneous because they can find a rigorous implementation in the context of a hyperreal number system, where a number with an infinite tail of $9$s can fall infinitesimally short of $1$ as outlined in a comment by user @GBeau on this page, namely if $H$ is an infinite hypernatural then $\displaystyle\sum_{n=1}^H \frac{ 9}{10} =0.999\ldots9$ where the digit $9$ occurs $H$ times.

This is of course a terminating infinite string of $9$s different from the one usually envisioned in real analysis, but it respects student intuitions and can be helpful in learning the calculus, as argued in Ely's fascinating study.

The existence of such an interpretation suggests that we indeed do assume that such a string represents a real number when we prove that it necessarily equals $1$.

Note I. If one thinks of the infinite string as being represented by the sequence $0.9, 0.99, 0.999, \ldots$ then one can obtain an alternative interpretation as follows. Instead of taking its limit (which is by definition real-valued), one can take what Terry Tao refers to as its ultralimit, to obtain a number than falls infinitesimally short of $1$.

These issues are dealt with in more detail in this recent publication.

The challenging philosophical issue here is the idea that there are distinct ways of formalizing infinity in mathematics, and the possibility of an attendant ambiguity of the symbol in question. These issues were dealt with in more detail in this publication in a leading education journal.

Note II. A certain number of objections have been raised by a colleague who wishes to remain anonymous. Given below are the objections together with my responses.

(0) You have not provided a meaningful syntactic representation of $1/3$ in the hyperreals.

Well $\dfrac13$ is the unending decimal $0.333\ldots$ (indexed by the hypernaturals). If truncated at infinite hypernatural rank $H$ this would produce a hyperrational falling infinitesimally short of a third, similarly to the $0.999\ldots{}$ situation.

(1) Nobody can legitimately disagree that hyperreals can be constructed via the ultraproduct of the reals $\bf{R}$ within $\sf{ZFC}$, which is the mainstream foundations for mathematics.

True, analysis with infinitesimals can be done over the hyperreals, as pointed out by Robinson in 1961. Alternatively, this can be done syntactically in the context of the ordinary real line, following Edward Nelson's approach. Nelson's approach, called Internal Set Theory $(\sf{IST})$, involves enriching the language of set theory by the introduction of a single-place predicate $\textbf{st}$, as well as three additional axiom schemas governing its interaction with the other set-theoretic axioms. Here $\textbf{st}(x)$ reads "$x$ is standard".

(2) Philosophically nobody has provided non-circular ontological arguments justifying $\sf{ZFC}$ (especially with replacement and choice). No logician, whether on Math SE or on Math Overflow or whom I have met, have done anything close to it.

This is a much broader issue. It is possible that $\sf{ZFC}$ has serious flaws. Nonetheless it happens to be currently the standard against which much of modern mathematics is tested. This doesn't mean that we must accept it, but it does mean that such philosophical problems are no smaller for the reals than for the hyperreals (especially in view of Nelson's syntactic approach mentioned above).

I accept various things such as consistency of $\sf{ZF}$ implying consistency of $\sf{ZFC}$, but consistency is quite irrelevant to soundness besides being necessary. Unless you're happy with just $\prod_1$-soundness.

If the sound alternative is predicativism as developed by Sol Feferman and others, then certainly $\sf{ZF}$ is no less problematic than $\sf{ZFC}$. Practically speaking, $\sf{ZF}$ is not enough for some rather standard applications. Consider the following example: it is consistent with $\sf{ZF}$ that there exists a strictly positive real function with vanishing Lebesgue integral; see https://arxiv.org/abs/1705.00493

(3) The construction of the hyperreals is via the ultraproduct of the reals R. If you can construct the hyperreals, then you also can construct $\bf{R}$ and prove the usual second-order real axioms for $\bf{R}$. It would be self-contradictory to say that the properties of $\bf{R}$ (including $0.999... = 1$ suitably interpreted) are not intuitive and then claim that the hyperreals are intuitive. After all, we define an infinitesimal in the hyperreals as a nonzero sequence of reals that converges to zero...

I wouldn't argue that the properties of the reals are not intuitive. Rather, what was explored in several articles in the recent literature is the possibility that there may be multiple approaches to interpreting the business with "a tail with an infinite number of $9$s", some of which may be helpful in harnessing student intuitions in a productive direction rather than merely declaring them to be erroneous.

Incidentally, your definition of a hyperreal infinitesimal is not quite correct.

An important distinction here is between procedures taught in a calculus class and set-theoretical justification (ontology of the entities involved) usually treated in an analysis course. This applies both to the reals and the hyperreals.

(4) Let $\bf{R}^\ast$ be the hyperreals and $\varepsilon = 1 - 0.999\ldots$. You claim that $\varepsilon$ is nonzero in a suitable interpretation of $0.999\ldots$ Ignoring the fact that you cannot represent $1/3$ meaningfully in similar decimal form, I now present you another fact that you can't represent $\varepsilon/2$, not to say $\sqrt{\varepsilon}$. Wait, what does the latter even mean in the hyperreals. Can your students figure that out? Are you sure hyperreals are so intuitive now?

I am not sure what you mean. Both $1/3$ and $\sqrt{\varepsilon}$ are well-defined in the hyperreals, simply by the transfer principle. As far as teaching the set-theoretic justification of the hyperreals in terms of the ultrapower, as I mentioned this belongs in a more advanced course, just like set-theoretic justification of the reals.

In contrast, asymptotic expansion can happily deal with $\sqrt{x}$ for any asymptotic expression $x$ that is non-negative. No trouble at all. $x^{1+x}$ for positive $x$? No problem.

All of these are well-defined over the hyperreals by the transfer principle.