> the decimal is a geometric series with a limit, right..
Right, but that's not really the crux of the matter. Hint: look at how the supremum is defined[1]. The definition of the supremum is how we end up with 0.999... = 1.
I suppose my point is that you could turn the repeating decimal into an infinite series, and a student might accept that, and you could define the suprememum and they might agree that it is 1, but then you ask them if the series is equal to the supremum, so they don't know what to do with the series, so they turn it into a sequence. Now they ask whether the last item in the sequence is equal to the supremum. Of course not! This is by definition.
And now you realize that you and the student have been operating by different rules. Their rules of equality are based on symbolic equality, so you actually have to relax the rules a bit to make limit equality work. And then, more importantly, you have to show that all the other rules are still intact. Actually, in this case, they aren't. Symbolic equality involving infinity is now horribly broken, and you have to express all equality in terms of limits to maintain consistency. Explore this further and you keep finding more inconsistencies that have to be settled by new rules that define new areas of mathematics.
So who is right? The natural world appears to be much more permissive than limit equality, preferring epsilon-equality. Symbolic equality is the only purely self-consistent system, but you can't do much with it. It's also possible that the natural world works with symbolic rules (quantum) but the complexity is great enough to resemble epsilon equality (continuum).
So, .999... == 1 by tautology. It's not some brilliant mathematical insight. The interesting part is the consequence of defining it as so.
And now you realize that you and the student have been operating by different rules. Their rules of equality are based on symbolic equality, so you actually have to relax the rules a bit to make limit equality work. And then, more importantly, you have to show that all the other rules are still intact. Actually, in this case, they aren't. Symbolic equality involving infinity is now horribly broken, and you have to express all equality in terms of limits to maintain consistency. Explore this further and you keep finding more inconsistencies that have to be settled by new rules that define new areas of mathematics.
So who is right? The natural world appears to be much more permissive than limit equality, preferring epsilon-equality. Symbolic equality is the only purely self-consistent system, but you can't do much with it. It's also possible that the natural world works with symbolic rules (quantum) but the complexity is great enough to resemble epsilon equality (continuum).
So, .999... == 1 by tautology. It's not some brilliant mathematical insight. The interesting part is the consequence of defining it as so.