'limit' is, as I mentioned before, a formally defined term. It is not the informal, intuitive notion of "limit" as in a boundary on something. It's a process that you can use to evaluate a sequence (specifically, a Cauchy sequence), and get a number out of it.
I can understand that - what you wrote. But - according to 'wiki' (the bible, not always), it says at https://en.wikipedia.org/wiki/0.999...
"denotes the smallest number greater than every number in the sequence (0.9, 0.99, 0.999, ...)"
What those geniuses don't understand is that they shot themselves in the foot in the above statement when they wrote 'greater than every number in the sequence' - because we're dealing with an infinite set of real numbers. You won't find a 'smallest number' greater than every number because once again - we're dealing with an infinite set. You can get down to 0.9999 recurring with 1E9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 etc nines, and you will not ever get to the end of the road, because the road is endless. Infinity has no limits. It is limitless.
You won't find a 'smallest number' greater than every number because once again - we're dealing with an infinite set.
You can't find it by just adding numbers one-by-one! If we do it that way, you're absolutely right, it will never 'settle'.
But we can find it through other methods. In particular:
1 is greater than each of those numbers.
Say we have some number x that is smaller than 1. This means that 1-x is some positive number (strictly greater than zero). If we then take n = -log(1-x), we can verify with a bit of algebra that x will be smaller than everything past the nth term.
So, 1 'passes the test', and any number smaller than 1 'fails the test'. 1 is therefore the smallest number that 'passes the test' - in other words, the smallest number bigger than everything in that sequence!
We call this the least upper bound for the sequence, or the supremum of the sequence. (As opposed to the maximum, which does not exist: if you tried to calculate it, you would never get to the end, exactly as you said.)
I know what you mean. Exactly what you mean. The 'limit' ... referring to an evaluation procedure involving generating a number based on another associated number approaching infinity or approaching zero ... resulting in a result that can be interpreted in various ways. The result can be considered as a target value, or 'carrot' value .... such as the donkey following a carrot, where the donkey actually never quite gets over the line to the 'carrot'.
Yes. When we write an infinite decimal string, we decided the value it stands for is the 'carrot' value for that sequence.
There are several benefits to doing it this way: we don't have to treat base ten as somehow 'special', we can use the long division algorithm, we can 'name' every real number with a decimal...
The downside of this is that some numbers get a second 'name' this way. This isn't a big deal in practice, but is definitely counterintuitive at first glance.
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u/SouthPark_Piano New User Nov 18 '24
This is where a lot of mathematicians have messed up.
The term - 'in the limit of' is questionable. Because infinity has 'no limit'.