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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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 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}?

In either case the answer is yes. Proof by contradiction:
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. Tags limit, number Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post zylo Calculus 13 May 31st, 2017 12:53 PM Shen Elementary Math 2 June 5th, 2014 07:50 AM date Calculus 3 June 12th, 2012 11:51 AM kiv864 Applied Math 0 November 2nd, 2010 05:38 PM Anson Complex Analysis 1 February 16th, 2010 04:25 PM

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