Hm?  I don't understand your meaning.  He accepts it as a convention, certainly.  The idea that it's accepted because not accepting it would mean the system needs an overhaul does not invalidate the first part in the least.  People are mostly bound by this system at this point and many don't think of it as a system but rather as reality itself.  That's wrong.

Some proofs that 0.999… = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference 1 − 0.999… must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean. Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses).^[48] A. H. Lightstone developed a decimal expansion for hyperreal numbers in (0, 1)^∗.^[49] Lightstone shows how to associate to each number a sequence of digits,

indexed by the hypernatural numbers. While he does not directly discuss 0.999…, he shows the real number ¹⁄₃ is represented by 0.333…;…333… which is a consequence of the transfer principle. As a consequence the number 0.999…;…999… = 1. With this type of decimal representation, not every expansion represents a number. In particular "0.333…;…000…" and "0.999…;…000…" do not correspond to any number.

The standard definition of the number 0.999… is the limit of the sequence 0.9, 0.99, 0.999, … A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, …)] of this sequence in theultrapower construction, which is a number that falls short of 1 by an infinitesimal amount. More generally, the hyperreal number u_H=0.999…;…999000…, with last digit 9 at infinite hypernatural rank H, satisfies a strict inequality u_H < 1.Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could be

All such interpretations of "0.999…" are infinitely close to 1. Ian Stewart characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999….^[51] Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about 0.999… < 1 are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.^[52][53] Jose Benardete in his book Infinity: An essay in metaphysics argues that some natural pre-mathematical intuitions cannot be expressed if one is limited to an overly restrictive number system:

There are other examples.  I don't think I need to paste them all.

Conegamer. Quick, someone light the Dog-Signal!

Nearly every single mathematical proof there is starts with stating something equivalent to "if x = 0.999... then:". If that is a valid reason to say my proof is invalid, then oh boy, we have to throw a lot of mathematics out the window! Also, the user I quoted was asking about the fraction proof, where I didn't use x at all.

Conjectures have been proven false but not so for mathematical proofs that have been verified by other mathematicians ... 

Using an infinite series to show that 0.999... = 1 is well established therefore OP is right and you wrong ... 

I have a degree in math, and I have a hard time accepting that one. I don't ever remember learning that, though.

Did you read Pokoko's post?

Basically theres math where 0.99999... = 1 doesnt make sense, and isnt true.

pokoko said: Faelco said:   And yet that Timothy Gowers explains well that refusing that "convention" would mean "invent strange new objects" or "abandon some of the familiar rules of arithmetic". So he clearly accepts it. Hm?  I don't understand your meaning.  He accepts it as a convention, certainly.  The idea that it's accepted because not accepting it would mean the system needs an overhaul does not invalidate the first part in the least.  People are mostly bound by this system at this point and many don't think of it as a system but rather as reality itself.  That's wrong. Some proofs that 0.999… = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference 1 − 0.999… must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same. However, there are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean. Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses).[48] A. H. Lightstone developed a decimal expansion for hyperreal numbers in (0, 1)∗.[49] Lightstone shows how to associate to each number a sequence of digits, indexed by the hypernatural numbers. While he does not directly discuss 0.999…, he shows the real number 1⁄3 is represented by 0.333…;…333… which is a consequence of the transfer principle. As a consequence the number 0.999…;…999… = 1. With this type of decimal representation, not every expansion represents a number. In particular "0.333…;…000…" and "0.999…;…000…" do not correspond to any number. The standard definition of the number 0.999… is the limit of the sequence 0.9, 0.99, 0.999, … A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, …)] of this sequence in theultrapower construction, which is a number that falls short of 1 by an infinitesimal amount. More generally, the hyperreal number uH=0.999…;…999000…, with last digit 9 at infinite hypernatural rank H, satisfies a strict inequality uH < 1.Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could be [50] All such interpretations of "0.999…" are infinitely close to 1. Ian Stewart characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999….[51] Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about 0.999… < 1 are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.[52][53] Jose Benardete in his book Infinity: An essay in metaphysics argues that some natural pre-mathematical intuitions cannot be expressed if one is limited to an overly restrictive number system: The intelligibility of the continuum has been found—many times over—to require that the domain of real numbers be enlarged to include infinitesimals. This enlarged domain may be styled the domain of continuum numbers. It will now be evident that .9999… does not equal 1 but falls infinitesimally short of it. I think that .9999… should indeed be admitted as a number … though not as a real number.[54] There are other examples.  I don't think I need to paste them all.

I edited my previous post. That quote and that paragraph are about alternative number systems, here "mathematically coherent ordered algebraic strutures, including various alternatives to the real numbers". Other examples are "mathematical structures with an addition operation but not a substraction operation".

It's not about the "classic" 1-9 and real "usual" numbers. Like it was said in the previous quote, these proves are done with other "artificial" alternative systems not working like the basic one, with other rules and operations.

And I'm not sure this discussion is about non-Archimedean number systems, but more about the "familiar rules of arithmetic".

We function in the reals.
0.999 is smaller than some epsilon, and 1 is in (1-epsilon, 1+epsilon) where epsilon gives us the "neighborhood" of all values close to 1. You initialized this in your first two lines where 10x=9.999.

When dealing with infinity, things can get weird. The 9s in 0.999... are infinite as are the 1s in .111...

This is worth a read: https://en.wikipedia.org/wiki/0.999...

I don't care about pokoko's post in this instance, what I care about is mathematics ...

It's your loss for not accepting the facts or the proofs ...