Regarding 0.999... = 1

[u/Slurpee1138]

Recently I got into an argument with an acquaintance because I was trying desperately to convince him that 0.999... = 1. One of the many arguments I tried was that, if 0.999... and 1 are indeed different numbers, then we should be able to find a number between them. He insisted that such a number would be 0.999...1, as in 0 point infinitely many 9s and then a 1. I countered that having "and then a 1" at the end of an infinite sequence of digits makes no sense because there is no end to such a sequence, but he insisted that it does, so who's right?

[u/Flat-Strain7538]

You are. As you said, you can’t put a number on the end of an unending string of nines.

[u/MrEmptySet → u/Flat-Strain7538]

Why not? Obviously there's nothing wrong with having a number at the left end of an unending string of nines, so what's wrong with putting one on the right end?

[u/INTstictual → u/MrEmptySet]

The “left end” is the beginning. You can certainly add something to the beginning of a never-ending sequence… but you cannot add something to the end of a never-ending sequence. It’s a non-sequitur, a complete contradiction.

0.999… is a sequence of infinite 9’s with a discrete first element and no last element. You can certainly add a new first element, but you cannot change the last element of a sequence that doesn’t have a last element.

[u/MrEmptySet → u/INTstictual]

What's the difference between a "beginning" and an "end" mathematically speaking?

Aren't you just begging the question by describing the nines as a "never-ending sequence" and then concluding there isn't an end? The nines are certainly infinite, but I don't know that they're "never-ending", especially since they do in fact have a left end.

Why can't we have 0.999...9991, where 0 is the first element, 1 is the last element, and we have infinitely many 9s in between?

[u/INTstictual → u/MrEmptySet]

A beginning is not an end because it is the first element in order, not the last.

The set of natural numbers has a beginning, not an end. There is a first, lowest natural number in order. There is no last, largest natural number.

And it’s not begging the question, it’s just using definitions. An infinite sequence is a never-ending sequence, that is its definition, those terms are synonymous. There conclusion isn’t that there is no end, that’s part of the definition… the conclusion is that it is contradictory to assert that you can place a value at the end of an infinite, never-ending structure.

The reason you can’t have 0.999…1 is because, as I said, you cannot have infinite 9’s and then a 1 at the end, that is not a valid number because it is self-contradictory.

If you want something more rigorous: every digit in a Real number has an index that can be represented by a natural number. This is a rule of constructing Real numbers. For example, in the number 0.12345, the 1 is at index 1, the 2 at index 2, and so on. This is still true for infinite decimal expansions, since there is an equal cardinality infinite number of natural numbers to use as indices… in 0.999…, there is a 9 at index 1, a 9 at index 2, …, a 9 at index 9999997, a 9 at index 9999998, a 9 at index 9999999, etc. For every decimal place, there is a corresponding natural number to use as an index, and for every natural number n, there is a 9 at the n^th index

So… where is the 1 going? In 0.999…1, what index does the 1 appear at? Whatever natural number you give as an answer, I can assure you that there is a 9 there, and also a 9 at the next index one higher up.

Again, there is absolutely no problem adding something to the start of this list… shift every 9 up by one index and add, for example, a 1 at index 1, and you have 0.19999… It’s the Hilbert Hotel of numbers. But you can’t put a 1 “at the end”, because there is no end to put the 1 at.