February 26th, 2018, 08:49 AM  #11  
Senior Member Joined: Oct 2009 Posts: 630 Thanks: 193  Quote:
$$S = \lim_n S_n$$ (whatever this means) For which exact natural number do we have $S=S_n$ then exactly?  
February 26th, 2018, 09:21 AM  #12  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote:
I prefer to think of Sn as "Something" n, where "Something" is defined for every (all) n, for example, a proposition, definition, nplace decimal, number sequence, Last edited by zylo; February 26th, 2018 at 09:29 AM.  
February 26th, 2018, 10:04 AM  #13  
Senior Member Joined: Oct 2009 Posts: 630 Thanks: 193  Quote:
So what would a natural number be if it has no exact explcit notation?  
February 26th, 2018, 10:23 AM  #14  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote:
I can define Sn for all n, so Sn has a definition for all n. So $\displaystyle \lim_{n \rightarrow \infty}$ defines S. If I could only define Sn for a finite number of terms, I couldn't define S. EDIT: Ex of infinite sequence of binary digit: 101010......10 to n places and let n $\displaystyle \rightarrow \infty$ Bottom line is, the only mathematical (hard) definition for $\displaystyle $\displaystyle \infty$$ I can think of is induction, ie, for all n. Just out of curiosity, what do you think "an infinite sequence of binary digits" means, and how would you give an example of one? Last edited by zylo; February 26th, 2018 at 10:39 AM.  
February 26th, 2018, 10:26 AM  #15 
Senior Member Joined: Oct 2009 Posts: 630 Thanks: 193 
Wait, are you talking about something like $$S=123456789101112....$$ and then an infinite number of digits? Your post seems to suggest this. You do realize that those things won't be natural numbers though, right? 
February 26th, 2018, 11:18 AM  #16  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote:
S is a specific, unique, unending sequence. As such, it is a member of {Sn} Last edited by zylo; February 26th, 2018 at 11:22 AM.  
February 26th, 2018, 11:48 AM  #17 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,512 Thanks: 2514 Math Focus: Mainly analysis and algebra  
February 26th, 2018, 12:17 PM  #18 
Senior Member Joined: Oct 2009 Posts: 630 Thanks: 193  
February 26th, 2018, 01:12 PM  #19 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
I said Sn was not a natural number. I simply meant it to be an example of something constructed, or defined by, natural numbers. If it makes it easier, put spaces between. S$\displaystyle _{11}$ = 1 2 3 4 5 6 7 8 9 10 11 $\displaystyle \lim_{n \rightarrow \infty}$ Sn =1 2 3,......,12578, .......... If none of {Sn} are unending, then there has to be a maximum value of n for Sn. There isn't. Last edited by skipjack; February 27th, 2018 at 01:37 AM. 
February 26th, 2018, 02:21 PM  #20  
Senior Member Joined: Aug 2012 Posts: 2,101 Thanks: 605  Quote:
I mention this so that you can calibrate the state of my knowledge. Can you please explain how $(S_n)$ has a different limit than $(n)$ in the hyperreals or hyperintegers? Thanks much. Last edited by Maschke; February 26th, 2018 at 02:33 PM.  

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