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October 19th, 2016, 07:18 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,546 Thanks: 110  Decimal Representation and Expansion
Definition: Property at infinity: P$\displaystyle _{\infty}$ =$\displaystyle \lim_{n\rightarrow \infty}$ P$\displaystyle _{n}$ You can't define a property at infinity without P$\displaystyle _{n}$. "Property at infinity" is convenient termijnology for the above definition; it does not mean a property evaluated at n=$\displaystyle \infty$, $\displaystyle \infty$ is not a number. Examples: 1) Decimal representation of numbers in [0,1). P$\displaystyle _{n}$=.499...9 P$\displaystyle _{\infty}$=.499....... 2) Decimal expansion of numbers in [0,1). P$\displaystyle _{n}$ = $\displaystyle \sum_{i=1}^{n}\frac{a_{i}}{10^{i}}$ .499...9 never equals .500...0, therefore .499.... never equals .500.... 3) Analytic definition of Limit. $\displaystyle \lim_{n\rightarrow \infty}$ .499.... = .500.... 4) Cantor's diagonal argument Notes .4999...9 means 4 followed by 9's to n places, and is representation or expansion depending on context. .499.... means 4 followed by an endless (infinite) string of 9's, and is representation or expansion depending on context. Last edited by skipjack; October 19th, 2016 at 08:43 PM. 
October 19th, 2016, 07:30 AM  #2 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,889 Thanks: 769 Math Focus: Wibbly wobbly timeywimey stuff.  Another one? (groans) Dan 
October 19th, 2016, 08:28 AM  #3 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,130 Thanks: 716 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
At this stage all I can recommend is getting a book on real analysis and then study the hell out of it, something I intend to do once I get free time (which unfortunately seems to be never for me )

October 19th, 2016, 10:30 AM  #4  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,445 Thanks: 2499 Math Focus: Mainly analysis and algebra  Quote:
Quote:
This is just as false as it was before you started trying to hide things behind your new terminology. Quote:
Cantor was certainly not using any of your terminology. He wasn't talking about any sort of limit either. He was talking about infinite sequences and enumerations. Last edited by skipjack; October 20th, 2016 at 12:02 PM.  
October 19th, 2016, 02:45 PM  #5 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,546 Thanks: 110 
Another example of the beauty, clarity, and simplicity of OP: What is $\displaystyle \infty^{2}$? $\displaystyle \infty^{2}$= P$\displaystyle _{\infty}$ where P$\displaystyle _{n}$ = n^{2} Without P$\displaystyle _{n}$, $\displaystyle \infty^{2}$ would be meaningless. 
October 19th, 2016, 03:02 PM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,445 Thanks: 2499 Math Focus: Mainly analysis and algebra 
It is meaningless. By your definition of $P_\infty$ doesn't exist. Again, there is no mechanism for an increasing $n$ to tell you anything about any infinity other than via the standard definition of convergent sequences. Your sequence doesn't converge. There are the ordinals that you could look at, where $\omega^2$ has some meaning. 
October 19th, 2016, 08:56 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 19,714 Thanks: 1806  
October 20th, 2016, 03:37 AM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,445 Thanks: 2499 Math Focus: Mainly analysis and algebra 
There a missing $\infty^2$ before the word "doesn't".
Last edited by v8archie; October 20th, 2016 at 03:40 AM. 
October 20th, 2016, 04:06 AM  #9 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,445 Thanks: 2499 Math Focus: Mainly analysis and algebra 
If we take Zylo's word that $\infty^2$ is meaningful, then his definition tells us that $\infty^2$ is finite (because every $n^2$ is finite). Specifically, there exists some natural number $m$ that is greater than $\infty^2$ (because for every $n^2$ there exists some natural number $m \gt n$). Moreover, we can say the same about $\infty$, so now "infinity" is finite and smaller than some natural number. 
October 20th, 2016, 08:21 AM  #10  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,546 Thanks: 110 
Reference Post#4 (with numbering added for convenience) Quote:
2) What is your definition of a property at infinity? I was simply clarifying an ambiguous terminology sometimes used incorrectly. 3) Taken out of context. 4) I didn't give definition of Analytic limit. I gave definition of Analytic limit of decimal expansion. 5) You can refer to an infinite sequence. But discussing its properties without P$\displaystyle _{n}$ is meaningless. Last edited by skipjack; October 20th, 2016 at 11:54 AM.  

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