
Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion 
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February 6th, 2019, 07:49 AM  #1 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  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. 
February 6th, 2019, 08:20 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,636 Thanks: 2621 Math Focus: Mainly analysis and algebra  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. 
February 6th, 2019, 04:51 PM  #3 
Senior Member Joined: Jun 2014 From: USA Posts: 505 Thanks: 39  
February 6th, 2019, 09:17 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,472 Thanks: 2039  
February 7th, 2019, 04:39 AM  #5 
Senior Member Joined: Jun 2014 From: USA Posts: 505 Thanks: 39 
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. 
February 7th, 2019, 06:09 AM  #6 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
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 goaround. 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. 
February 7th, 2019, 06:48 AM  #7  
Senior Member Joined: Jun 2014 From: USA Posts: 505 Thanks: 39  Quote:
Quote:
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 $n,m \in \mathbb{N}$.  
February 7th, 2019, 07:52 AM  #8  
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  Quote:
No one that I can recollect has disputed that, in the ideal world of mathematics, a 1to1 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 1to1 correspondence with the set of natural numbers starting with 1 and ending with 10^n. But that is irrelevant to infinite strings.  
February 7th, 2019, 08:24 AM  #9  
Global Moderator Joined: Dec 2006 Posts: 20,472 Thanks: 2039  Quote:
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.  
February 7th, 2019, 11:19 AM  #10 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
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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limit, natural, number, series 
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