Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?
You might consider "real" numbers (in plain English) to mean physically measurable quantities, but there are plenty of numbers that we can write out because they're infinitely long, but trivially "made" (such as π, which just requires grabbing a compass and drawing a circle)
The bit you should be wondering about is why 0.666...6 and 0.666...7 are the same number: infinities cause digits written on paper (or a computer screen) to look like a kind of number that they're not. The two fractions (numbers in ℚ) 0.6666 and 0.66667 are 0.00001 apart, but the two reals (numbers in ℝ) 0.666...6 and 0.666...7 are 0.000...1 apart. That looks like a tiny tiny fraction, but it's not a fraction, it's an infinite number of zeroes, and thanks to that, this number, while it looks like a fraction, is just a silly way to write zero.
So thanks to infinities, the most-definitely-not-a-fraction number that we write as 0.666... is the same as the most-definitely-not-a-fraction number 0.666...(some numbers here). The difference between the two is zero.
Infinities are fun. And difficult. But also fun.
Ill-defined. Try again.
The blog post that (I think) started off this chain of posts, at https://blog.plover.com/misc/half-baked.html , has the most concise and lucid explanation I've found, so I'm just going to straight-up copy it: 0.666...7 is an “an object... said to ‘have order type ω+1’, and is completely legitimate.”
It's not very useful – it's exactly equal to 0.666... ! – but it's legitimate and well-defined.
My absolute favorite construction of objects like this is Conway's surreal numbers. These things appear perfectly naturally in the surreal numbers, and are completely well-defined, if (again) not very useful.
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1. we assert that 0.666...7 is a sequence of digits.
2. we assert that each digit in a sequence can be assigned an integer index corresponding to its position in the sequence. I.e. we can defined each digit's index as "the number of digits that precede this digit".
3. from (1) and (2) it follows that the index for 7 must be an integer.
4. the ellipses represents an infinite number of digits (infinitely repeating the repeated digit pattern preceding it).
5. from (2) and (4) it follows that the index for 7 must be the integer value "infinity", because it has an infinite number of digits preceding it.
6. (3) and (5) cannot both be true, because infinity is not an integer.
7. from (6) it follows that (1), and/or (2), and/or (4) must be false
8. (4) is, by definition, true.
9. from (7) and (8) it follows that (1) and/or (2) must be false.
10. (1) is our fundamental assertion. If (1) was false then there is wouldn't even be a sequence of digits for us to reason about. So (1) is true.
11. from (6), (8), and (10) it follows that (2) must be false for there to be no contradiction.
QED
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Now, certainly, for finite length numbers the assumption that each digit in a sequence has an integer index holds true, but it turns out we have mathematical notation that lets us write down numbers for which that property does not hold.
Infinities are fun. And difficult. But also fun.
A sequence in the mathematical sense is a function whose domain is the natural numbers. Please define that function for the creature you're working with here. Otherwise you're trying to prove things about an object with no definition. You will end up in trouble.
As it currently stands, assumption (1) is similar in nature to me saying "gnarfgnarf is an imaginary number". It's completely meaningles unless I define what I mean by gnarfgnarf.
So: what do you mean by 0.666…7?
Same here: we have a number written as 0.666...7 using conventional mathematical notation. The comment that is being replied to asserts that this can be treated as a sequence, and so we start the proof with that definition: "0.666...7 is a sequence", and now we're done. You, as reader of the proof, have been informed that those nine symbols, in that order, for the rest the proof, represent a sequence. Not "a specific sequence", but "any sequence", and it must follow all the rules that sequences follow.
We then show that simply by being "a sequence", due to the properties of sequences, we get a contradiction. Our first assertion is the definition for the purpose of this proof, and is sufficient.
If someone says 0.666…7 then you have a couple different ways you can take the discussion. You can say, “No! Real numbers don’t work like that!” or you can talk about what number systems would look like if you can do that.
It turns out that there’s a lot to learn from the alternative number systems, including formulations of calculus without limits that are easier to understand from an intuitive perspective, yet equally rigorous. The field is called “nonstandard analysis”. It’s not taught in college.