My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Thanks Tree14Thanks
Reply
 
LinkBack Thread Tools Display Modes
February 24th, 2018, 08:50 AM   #1
Senior Member
 
Joined: Mar 2015
From: New Jersey

Posts: 1,364
Thanks: 100

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
zylo is offline  
 
February 24th, 2018, 09:48 AM   #2
Senior Member
 
romsek's Avatar
 
Joined: Sep 2015
From: USA

Posts: 1,932
Thanks: 1000

This doesn't really make any sense as written.

what is $S_{10}$ ?

Are you supposing the existence of infinite digits?
romsek is online now  
February 24th, 2018, 01:36 PM   #3
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 377
Thanks: 205

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.
SDK is offline  
February 24th, 2018, 02:08 PM   #4
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,305
Thanks: 2443

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.
v8archie is offline  
February 25th, 2018, 12:14 PM   #5
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 3,159
Thanks: 866

Quote:
Originally Posted by v8archie View Post
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 View Post
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.
Country Boy is offline  
February 25th, 2018, 12:57 PM   #6
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,305
Thanks: 2443

Math Focus: Mainly analysis and algebra
Er... Yeah. Ooops.
v8archie is offline  
February 25th, 2018, 01:38 PM   #7
Senior Member
 
Joined: Aug 2012

Posts: 1,887
Thanks: 525

Quote:
Originally Posted by SDK View Post
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.
Maschke is offline  
February 26th, 2018, 06:27 AM   #8
Senior Member
 
Joined: Mar 2015
From: New Jersey

Posts: 1,364
Thanks: 100

Quote:
Originally Posted by Maschke View Post
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.
zylo is offline  
February 26th, 2018, 06:51 AM   #9
Senior Member
 
Joined: Oct 2009

Posts: 402
Thanks: 139

Quote:
Originally Posted by Maschke View Post
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 $n$.
Micrm@ss is offline  
February 26th, 2018, 07:30 AM   #10
Senior Member
 
Joined: Mar 2015
From: New Jersey

Posts: 1,364
Thanks: 100

Quote:
Originally Posted by zylo View Post
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.
zylo is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
limit, number



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
upper limit = lower limit implies convergence zylo Calculus 13 May 31st, 2017 12:53 PM
natural number multiple of another number if its digit sum equal to that number Shen Elementary Math 2 June 5th, 2014 07:50 AM
Evaluation of error in limit of number e date Calculus 3 June 12th, 2012 11:51 AM
LIMIT ANALYTHIC OR ROTATION NUMBER? kiv864 Applied Math 0 November 2nd, 2010 05:38 PM
how to proof sequence limit for complex number Anson Complex Analysis 1 February 16th, 2010 04:25 PM





Copyright © 2018 My Math Forum. All rights reserved.