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February 6th, 2019, 08:49 AM   #1
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Limit of a Natural Number Series

Limit of a Natural Number Series

Take a line marked off in unit intervals: 0,1,2,.....

Pick a point in (0,1)

Divide [0,1] in ten intervals and say p is in fifth interval. Write .4 and mark 4 on the line.

Divide fifth interval in 10 again and say p is in seventh sub interval. Write .46 and mark 4+6 =10 on the line.

See where this is going? Both sequences approach a definite point on the line.
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February 6th, 2019, 09:20 AM   #2
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Quote:
Originally Posted by zylo View Post
Both sequences approach a definite point on the line.
Stated without proof.

It's trivial to see that your "sum of natural number" series does not converge unless it happens to be a finite sequence - i.e. you have marked a point in (0,1) that represents a terminating decimal.

Otherwise, infinitely many terms of the infinite series are greater than 1, meaning that the sequence of terms cannot tend to zero which, as anybody who has studied infinite series knows, indicates that the series cannot converge. Instead it grows without bound.
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February 6th, 2019, 05:51 PM   #3
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Originally Posted by zylo View Post

See where this is going?
Yup. To infinity. Or nowhere. Or both.
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February 6th, 2019, 10:17 PM   #4
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Quote:
Originally Posted by zylo View Post
Both sequences approach a definite point on the line.
The second sequence is geometrically undefined, as you haven't stated where "10", for example, is marked on the line (other than for the original unit interval markings, which seem to be irrelevant).
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February 7th, 2019, 05:39 AM   #5
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It seems that $0.0\overline{1}$ and $0.00\overline{1}$ would both result in all the whole numbers being marked. Any number of the form $0.\text{[finite number of 0's]}\overline{1}$ would.

$0.0\overline{1}$ would mark: 0, 0+1, 0+1+1, 0+1+1+1, ... = 0, 1, 2, 3, ...
$0.00\overline{1}$ would mark: 0, 0+0, 0+0+1, 0+0+1+1, 0+0+1+1+1, ... = 0, 0, 1, 2, 3, ...

Hey zylo, is there any other way two real numbers would result in the same sequence of whole numbers being marked?

If so, are there an infinite number of real numbers that would end up marking the same sequence of whole numbers for some, or for all, sequences of whole numbers that could be marked using your method?

Does that help you see where this is going? Even if I allowed you the false notion that a steadily increasing infinite sequence of whole numbers approached a definite point on the line (which only happens in zylo world where infinite integers with magical properties ride unicorns off into the sunset every night), you still have much work to do.

...Or, you could just post something to the effect of how you take too many drugs and you were as high as a kite for this one.
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February 7th, 2019, 07:09 AM   #6
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Very perceptive Aplanis. That's why I said elsewhere you could count the real numbers in [.1,1) by removing the decimal point. Dealing with [0,1) requires an artifice and I like to keep things clean for a first go-around.
In any event, it's not necessary in this thread because a sum of natural numbers begins with 1, not 0.

While I'm at it, note that if a point lies on an interval, so that you have a finite number n of decimal places, say, .145, The sum 1+4+5 certainly exists as a specific integral point on the line, for any n.
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February 7th, 2019, 07:48 AM   #7
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Quote:
Originally Posted by zylo View Post
Very perceptive Aplanis. That's why I said elsewhere you could count the real numbers in [.1,1) by removing the decimal point. Dealing with [0,1) requires an artifice and I like to keep things clean for a first go-around.
In any event, it's not necessary in this thread because a sum of natural numbers begins with 1, not 0.
Going with the later option and taking the high road I see:

Quote:
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...Or, you could just post something to the effect of how you take too many drugs and you were as high as a kite for this one.
How about a good "while you're at it" tidbit?

Quote:
Originally Posted by zylo View Post
While I'm at it, note that if a point lies on an interval, so that you have a finite number n of decimal places, say, .145, The sum 1+4+5 certainly exists as a specific integral point on the line, for any n.
That'll do. Given your example and using your method, 0.1045 would result in the same sum = 1+0+4+5 = 10. You really like terminating decimals it seems. We keep somehow getting back to you noticing those numbers of the form $\frac{n}{10^m}$, where $n,m \in \mathbb{N}$.
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February 7th, 2019, 08:52 AM   #8
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Quote:
Originally Posted by zylo View Post
Very perceptive Aplanis. That's why I said elsewhere you could count the real numbers in [.1,1) by removing the decimal point. Dealing with [0,1) requires an artifice and I like to keep things clean for a first go-around.
In any event, it's not necessary in this thread because a sum of natural numbers begins with 1, not 0.

While I'm at it, note that if a point lies on an interval, so that you have a finite number n of decimal places, say, .145, The sum 1+4+5 certainly exists as a specific integral point on the line, for any n.
You truly do not pay any attention to what others say.

No one that I can recollect has disputed that, in the ideal world of mathematics, a 1-to-1 correspondence can be set up between the real numbers in the interval of (0, 1) and a set of carefully defined strings, each of which has a countably infinite number of decimal digits, whether or not that string is preceded by 0 and a decimal point.

What has been disputed is that it is generally meaningful to talk about constructing an endless string of digits by starting with the end digit of a different endless string of digits.

What I also recollect people doing is disputing the legitimacy of a vocabulary such that you can talk about counting the real numbers in an interval.

No doubt you may stipulate definitions that are useful to an argument, but you must disclose such definitions when they differ from what are generally accepted definitions and also must be consistent thereafter in using such stipulated definitions. But you do not do this.

For example, either you are using the word "counting" in a sense that is not standard in the field of analysis, or you are assuming what you wish to prove.

Finally, no one has disputed that it is possible to place the set of all possible strings of decimal digits that are of length n, where n is a natural number, into 1-to-1 correspondence with the set of natural numbers starting with 1 and ending with 10^n. But that is irrelevant to infinite strings.
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February 7th, 2019, 09:24 AM   #9
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Originally Posted by zylo View Post
I said elsewhere . . . I like to keep things clean for a first go-around
That's equivalent to stating "I'm repeating myself", yet you are still making no attempt to state your assumptions and define your terminology, instead relying on not doing so and not replying to key observations about what you post.

Quote:
Originally Posted by zylo View Post
See where this is going? Both sequences approach a definite point on the line.
It's okay to "keep things clean" by just specifying the definite point and why it's approached. You're apparently being vague just for the sake of it.
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February 7th, 2019, 12:19 PM   #10
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Don't agonize over it if you didn't get the point right away. It's not that tight a post.

There are n! natural number corresponding to n place digits, FOR ALL n.

There seems to be this notion floating around that if a sequence of digits is very large it can't be a natural number.

There is no ONE natural number at "infinity."
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