No problem. It is a great proof (there aren't many so simple and succinct). Just bad luck on timing ;)

Yes, I totally agree with you: consistency and convenience are why we have chosen to use 1.9999... notation to represent the limit, rather than the sequence.

consistency and convenience tends to drive most mathematical notational choices (with occasional other influences), for reasons that should be extremely obvious.

It just so happened that, o this occasion, I was not aware enough of either the actual convention, or of other "things that this notation would be consistent with" before I guessed at the meaning of this particular item of notation.

And so my guessed meaning was one of the two things that I thought would be "likely meanings" for the notation.

In this case, my guess was for the wrong one of the two.

I seem to be getting a lot of comments that are implying that I should have somehow naturally realised which of the two meanings was "correct"... and have tried very hard to explain why it is not obvious, and not somehow inevitable.

Both of my possible interpretations were potentially valid, and I'd like to insist that the sequence-one is wrong only by convention (ie maths has to pick one or the other meaning... it happens to be the most convenient for mathematicians, which happens in this case to be the limit-interpretation)... but as is clearly evidenced by the fact that there is so much confusion around the subject (ref the wikipedia page) - it is not obvious intuitively that one is "correct" and one is "not correct".

I maintain that without knowledge of the convention, you cannot know which is the "correct" interpretation. Any assumption otherwise is simply hindsight bias.

is "x-squared = 2" ? is a perfectly valid question to ask in mathematics even though the LHS is not obviously an number

In this case, it is a formula that can equate to a number... just as the sequence is a (very limited) formula that can equate to 2 - if we take the sequence to its limit; or that falls just shy of 2 - if we try and represent it in any finite/limited way.

In stating that 1.9999... is a number, you are assuming the usage of the limit/number, rather than the other potential usage ie, you are falling into the same assumption-trap that I fell into... It's just that your assumption happens to be the one that matches with common usage, whereas mine wasn't ;)

Using 1.9999. to represent the limit of the sequence (ie the number) is certainly true by convention (ie "by definition"), but is no means the only way to interpret the symbols. It could just as easily represent the sequence itself... we just don't happen to do that - we define what mathematical symbols refer to... they're just the word/pointers to what we're talking about yes?

Hi Misha, I might also turn that argument back on you and repeat what I said before: "if you meant 2, why not just say 2?" It's as valid as "if you meant the sequence, why not just write {1, 1.9, 1.99, 1.999, ...}"?

Clearly there are other reasons for using something that is not the usual convention. There are definitely good reasons for representing infinite series or sequences... as you have pointed out. However - there is no particular reason why mathematics has chosen to use 1.999... to mean the limit, as opposed to the actual infinite series. Either one could be equally validly used in this situation.

It is only by common convention that mathematics uses it to represent the actual limit (as n tends to infinity) instead of the other possibility - which would be "the actual limit as n tends to infinity... if we actually take it to infinity, or an infinitesimal less than the limit if we don't", which is how I assumed (incorrectly) that it was to be used

However, the other thing you say that "we never denote it 1.999..." pulls out an interesting though, and if I grasp what you're saying correctly, then I disagree with you.

As I've mentioned in another comment now - mathematical symbolic conventions are the same as "words" - they are map, not territory. We define them to mean what we want them to mean. We choose what they mean by common consensus (motivated by convenience). It is a very good idea to follow that convention - which is why I decided I was wrong to use it the way I originally assumed it was being used... and from now on, I will use the usual convention...

However, you seem to be saying that you think the current way is "the one true way" and that the other way is not valid at all... ie that "we would never denote it 1.9999..." as being some sort of basis of fact out there in reality, when really it's just a convention that we've chosen, and is therefore non-obvious from looking at the symbol without the prior knowledge of the convention (as I did).

I am trying to explain that this is not the case - without knowing the convention, either meaning is valid... it's only having now been shown the convention that I now know what is generally "by definition" meant by the symbol, and it happened to be a different way to what I automatically picked. without prior knowledge.

so yes, I think we would never denote the sequence as 1.999... but not because the sequence is not representable by 1.999... - simply because it is conventional to do so.