This little guy craves the light of knowledge and wants to know why 0.999... = 1. He wants rigour, but he does accept proofs starting with any sort of premise.
Enlighten him.
this post was submitted on 02 Jul 2024
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chapotraphouse
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The thing that really got 0.999...=1 to click on my mind is the fact that you can't find a number between 0.999... and 1. You might think "Just put something at the end of 0.999...," but there is no end to 0.999....
Yay, Dedekind cuts are finally useful!
I'm not sure what you are trying to say here, and I have a background in math. I think this is just going to confuse lay people.
Im trying to make the distinction between a function that approaches a value as it's input grows, for instance a sequence seen as a function on the domain of the natural numbers, and the value itself.
I have seen a lot of people view 0.999... as a number that "approaches one", so formally speaking as the sequence (0.9, 0.99, 0.999, ...) and not the number itself which that sequence approaches which they would agree is 1.
The "its the largest number which is less than 1" type of thinking.
I am going to note that this was not well-expressed when you said 'we can just pretend to have "reached infinity" and work with like any number'. To a lay person it would look as if you were suggesting that we non-rigorously treat one object (like the sequence (0.9, 0.99, 0.999,...)) as another (like the real number that that sequence converges to given the standard topology of the space of real numbers).
I'm not really confused about what you're saying here exactly, and since the original post is deleted, I can't really even see what was originally said, but I was confused about this:
Why make mention of the standard topology here exactly? It's not exactly clear to me why this has anything to do with what you two are discussing.
Just to be specific, as what a particular sequence converges to depends on the topology of the space where we are looking for a limit of the sequence. Hell, in non-Hausdorff spaces a sequence can have multiple limits (trivial case: anti-discrete space of cardinality greater than 1 will have every sequence converge to every point in it).
Thanks for the clarification! In my mind, I sort of just think "metric first" so the topology induced by that metric is always just assumed, but that's because I don't ever work with non-metrizable spaces.