Nope, not doing this again.
Step #1 is in error already. The limit of (0.9, 0.99, 0.999, …) is 1. 0.999… is not the limit since it is not a point on the number line. 0.999… is greater than every element in { 0.9, 0.99, 0.999, … }, but you cannot find a segment in 0.999… which is not already a segment of one of the numbers 0.9, 0.99, 0.999, …
So, after all this time, the key point is that you refuse to accept 0.9999 as an alternate notation for 1.0000. It might have saved some time if you’d made that point explicit earlier.
by the same logic, do you not accept 0.3333 an an alternate notation for 1/3 ?
The limit of (0.3, 0.33, 0.333, …) is ⅓. 0.333… is not the limit since it is not a point on the number line. 0.333… is greater than every element in { 0.3, 0.33, 0.333, … }, but you cannot find a segment in 0.333… which is not already a segment of one of the numbers 0.3, 0.33, 0.333, …
So “number”, I guess, means real number. But only a tiny subset of the rationals (those with only 2 and 5 as prime factors in the denominator) are “points on the number line.”
Got it.
Now that that’s settled, can you point us to the scientific fields where this new “insight” is useful? String theory? Computational geometry?
We are dealing with simple logic. That’s it.
0.9
0.99
0.999
…
can be written as
[0, 0.9]
[0, 0.9]∪[0.9, 0,99]
[0, 0.9]∪[0.9, 0,99]∪[0.99, 0,999]
…
0.999… can be written as [0, 0.9]∪[0.9, 0,99]∪[0.99, 0,999]∪…
There is no segment in 0.999… which is not already a segment of one of the finite numbers in the list.
Usually we are saying that a number is greater than a set of smaller numbers, if it contains at least one segment which is not a segment of one of the smaller numbers.
Yes, the simple logic of a 4-year-old who can’t figure out that the sun is remaining in the same place relative to the earth, and merely appears to be moving because the earth is spinning. “But we haven’t moved, it’s moving!”
That’s not what I described at all. I guess you’re just not able to understand what we’re saying. No point in continuing.
Please, don’t randomly change the equation as given … I asked about the limit of the summation … and I’m assuming here you agree that the limit of the summation as n –> ∞ is in fact 1 …
Thank You …
So, it looks like the issue here is that you don’t believe that for most functions, f(x) … the limit as x –> a of f(x) is equal to f(a) … For example if g(x) = 2x + 3 … then the limit as x –> 4 of g(x) is equal to 11 … sure as shit, g(4) is also equal to 11 … there are exceptions to this … for h(x) = 1/x (where x ≠ 0) … the limit as x –> 0 is equal to infinity … however h(0) is undefined, since x ≠ 0 … but that’s not the case in the summation we’re working with here … all the terms are clearly defined …
So let me ask another question … why is it you don’t believe us when we say “the limit as x –> a of f(x) is equal to f(a) in this case”? We can prove this to you, but you’re going to need two years college calculus under your belt … no way can we teach you calculus here on these boards … no way can we develop the theorems needed which takes a full 24 units of college credit …
At some point here … you’re going to have to call us liars … millions upon millions of mathematicians all conspiring just to deceive you … and only you … and all seven billion of us in on the deception … just to deceive you … and only you … James Maxwell, Albert Einstein, Stephen Hawking … their entire body of work all designed to deceive you … and only you …
[sigh]
You’re ending your logic too soon … giving you an incomplete conclusion … only a half truth … one that fails in the real world …
And yet, as a whole, 0.99… goes beyond those finite numbers. Next thing, you’re going to be claiming that Gabriel’s Horn couldn’t possibly have finite volume, because it has infinite area. You don’t have the slightest clue what you’re talking about and your “logic” was discarded as useless and/or meaningless more than 2000 years ago.
Trouble with logic is that if your premises are false, your conclusion cannot be trusted.
All horses are female.
‘Dobbin’ is a horse.
Dobbin is female.
The logic is valid.
OK, let’s review basic logic. We’ll start by proving the well-known fact that all horses have an infinite number of legs.
Casinos ban winning gambling systems … casinos ban throwing the dice as hard as one can … therefore, throwing the dice as hard as one can is a winning gambling system …
I’ve actually tested this to some degree, usually the dealers don’t say anything until someone gets hurt … but every time I get the dice to bounce off both end walls … they come up a winner …
[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… means 0.999… as a whole.
I bet you cannot name a segment in 0.999… which is not a segment of one of the numbers of the list. And yet, you are claiming that there is.
Not my claim.
No one has shown yet how we reach the limit point in my previous examples.
[0.9, 1], because 0.99… = 1, because there is no distance you can point to between 0.99… and 1.
Done.
That was easy.
No, but it’s exactly the same kind of intuitive, ultimately faulty logic.
Your previous examples required we treat infinity as a very very large finite number … which is untrue … and the limits of your education don’t allow for you to understand what you have been shown … please check with your mom or dad, see if either still have their old calculus textbook you can borrow … we can try to explain a simplistic epsilon/delta proof (again).
[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… doesn’t include 1 as an endpoint.
[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1). Try again.
Even in my “copy and paste” examples I am dealing with infinity which is completed in a finite amount of time. So, I definitely agree that Gabriel’s Horn has finite volume although it has infinite area.