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 February 24th, 2018, 08:50 AM #1 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Limit of a Number S = $\displaystyle \lim_{n\rightarrow \infty}$ 123....n $\displaystyle S_{n}$ = 123....n for all n. Is S a member of {$\displaystyle S_{n}$}? Why? Last edited by zylo; February 24th, 2018 at 08:56 AM. Reason: Why? added
 February 24th, 2018, 09:48 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,552 Thanks: 1402 This doesn't really make any sense as written. what is $S_{10}$ ? Are you supposing the existence of infinite digits?
 February 24th, 2018, 01:36 PM #3 Senior Member   Joined: Sep 2016 From: USA Posts: 645 Thanks: 407 Math Focus: Dynamical systems, analytic function theory, numerics If I understand this correctly, then $S_{10}$ would be the integer 12345678910. If so, the limit doesn't exist, so claiming $S$ is equal to it is meaningless, as is asking whether or not the limit is a member of any set. Thanks from topsquark and v8archie Last edited by skipjack; February 27th, 2018 at 12:19 AM.
 February 24th, 2018, 02:08 PM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2664 Math Focus: Mainly analysis and algebra We seem to have all sorts of problems here. Not least that a number doesn't have a limit. Functions and sequences have limits (indeed sequences are functions with the natural numbers as their range). Limits aren't defined for any other objects that I can think of right now. SDK has hit the nail on the head for the implied sequence in your limit. Thanks from topsquark Last edited by skipjack; February 27th, 2018 at 12:15 AM.
February 25th, 2018, 12:14 PM   #5
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Quote:
 Originally Posted by v8archie We seem to have all sorts of problems here. Not least that a number doesn't have a limit. Functions and sequences have limits (indeed sequences are functions with the natural numbers as their range).
You mean "as their domain".

Quote:
 Originally Posted by v8archie Limits aren't defined for any other objects that I can think of right now. SDK has hit the nail on the head for the implied sequence in your limit.

Last edited by skipjack; February 27th, 2018 at 12:23 AM.

 February 25th, 2018, 12:57 PM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,683 Thanks: 2664 Math Focus: Mainly analysis and algebra Er... Yeah. Ooops.
February 25th, 2018, 01:38 PM   #7
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Quote:
 Originally Posted by SDK If I understand this correctly, then $S_{10}$ would be the integer 12345678910. If so, the limit doesn't exist, so claiming $S$ is equal to it is meaningless, as is asking whether or not the limit is a member of any set.
It's perfectly sensible to say that $\displaystyle S = \lim_{n \to \infty} S_n = + \infty$ in the extended real number system. And that $S \notin \{S_n\}_{n \in \mathbb N}$.

In fact, $(S_n)$ is a subsequence of $(n) = 1, 2, 3, 4, 5, \dots$, which has the same limit.

Last edited by skipjack; February 27th, 2018 at 12:21 AM.

February 26th, 2018, 06:27 AM   #8
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Quote:
 Originally Posted by Maschke It's perfectly sensible to say that $\displaystyle S = \lim_{n \to \infty} S_n = + \infty$ in the extended real number system. And that $S \notin \{S_n\}_{n \in \mathbb N}$. In fact $(S_n)$ is a subsequence of $(n) = 1, 2, 3, 4, 5, \dots$, which has the same limit.

And that $S \notin \{S_n\}_{n \in \mathbb N}$ Why?
===========================

Sn, by definition, is a construction (artificial).
If Sn is 12345.....n, what else could Sn be but 12345,....,10. If it's clearer, put a space between the numbers

It's no different conceptually (see Maschke above) than Sn=n, for all n, and S=$\displaystyle \lim_{n \rightarrow \infty}$. Is S in {Sn}?

Assume S is not in {Sn}. Then there is an Sn for which n is a maximum. Contradiction. There is no maximum for n.

It's really induction in disquise. If you don't accept induction as a definition of infinity, then "infinity" is not a mathematical term. If you don't precisely define your terms, it becomes a political discussion.

For all (every) n iff induction defines infinity.

February 26th, 2018, 06:51 AM   #9
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Quote:
 Originally Posted by Maschke It's perfectly sensible to say that $\displaystyle S = \lim_{n \to \infty} S_n = + \infty$ in the extended real number system. And that $S \notin \{S_n\}_{n \in \mathbb N}$. In fact $(S_n)$ is a subsequence of $(n) = 1, 2, 3, 4, 5, \dots$, which has the same limit.
True, but I wonder if the extended real numbers are the best setting to see this.
I think it is worth checking out the hyperreals. The hyperreals are by definitions all the sequences in $\mathbb{R}$ modulo some equivalence relation.

In this system, $S_n$ does not have the same limit as $n$.

February 26th, 2018, 07:30 AM   #10
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Quote:
 Originally Posted by zylo S = $\displaystyle \lim_{n\rightarrow \infty}$ 123....n $\displaystyle S_{n}$ = 123....n for all n. Is S a member of {$\displaystyle S_{n}$}? Why?
Or, if you prefer, Sn=n.

What does the extended number system, a definition, or hyperreals have to do with this?

EDIT
As I wrote in a previous post, my answer is yes. Assume S is not in {Sn}. Then there is an Sn for which n is a maximum. Contradiction. There is no maximum for n.

Last edited by zylo; February 26th, 2018 at 07:39 AM.

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