Register to reply

Negation of limit definition

by antiemptyv
Tags: definition, limit, negation
Share this thread:
antiemptyv
#1
Sep20-07, 02:22 PM
P: 34
1. The problem statement, all variables and given/known data

I'm trying to show that a sequence does not have a limit, so that would mean proving the negation of the limit definition is true, right? Is this a correct negation of the definition of what it means for a sequence to have a limit?

2. Relevant equations

The definition of the limit of a sequence [tex](x_n)[/tex].
The sequence [tex](x_n)[/tex] converges to [tex]L[/tex] if given [tex]\epsilon > 0[/tex], [tex]\exists K(e) \in \mathbb{N} \ni[/tex] if [tex]n > K(e)[/tex], then [tex]|x_n-L| < \epsilon[/tex].

3. The attempt at a solution

The limit of a sequence [tex](x_n)[/tex] is not L if [tex]\exists \epsilon > 0 \ni \forall K \in \mathbb{N}[/tex], [tex]\existsn \in \mathbb{N} \ni n > K \ni |x_n - L| \geq \epsilon[/tex].
Phys.Org News Partner Science news on Phys.org
NASA team lays plans to observe new worlds
IHEP in China has ambitions for Higgs factory
Spinach could lead to alternative energy more powerful than Popeye
EnumaElish
#2
Sep20-07, 02:39 PM
Sci Advisor
HW Helper
EnumaElish's Avatar
P: 2,483
I think that is right, except it seems as if you have used too many N's, \in's or \ni's.
HallsofIvy
#3
Sep20-07, 03:49 PM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,310
Quote Quote by antiemptyv View Post
1. The problem statement, all variables and given/known data

I'm trying to show that a sequence does not have a limit, so that would mean proving the negation of the limit definition is true, right? Is this a correct negation of the definition of what it means for a sequence to have a limit?

2. Relevant equations

The definition of the limit of a sequence [tex](x_n)[/tex].
The sequence [tex](x_n)[/tex] converges to [tex]L[/tex] if given [tex]\epsilon > 0[/tex], [tex]\exists K(e) \in \mathbb{N} \ni[/tex] if [tex]n > K(e)[/tex], then [tex]|x_n-L| < \epsilon[/tex].

3. The attempt at a solution

The limit of a sequence [tex](x_n)[/tex] is not L if [tex]\exists \epsilon > 0 \ni \forall K \in \mathbb{N}[/tex], [tex]\existsn \in \mathbb{N} \ni n > K \ni |x_n - L| \geq \epsilon[/tex].
Not if by "if [itex]n> K(e)[/tex] then [tex]|x_n_L|< \epsilon[/itex] you mean "for all n> N(e).
That only has to be true for some n> Ke)

antiemptyv
#4
Sep20-07, 08:17 PM
P: 34
Negation of limit definition

Yes, it all seems right now I guess. Thanks! and oh yeah, i guess while editting, i left in a few extra symbols...


Register to reply

Related Discussions
Negation of Limit Calculus & Beyond Homework 5
Definition of a Limit. Calculus & Beyond Homework 3
Definition of a Limit. Calculus & Beyond Homework 0
The definition of limit Calculus 8
Use definition of limit... Calculus 4