Is the Converse of the Sequence Converging Proof True?

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Homework Help Overview

The problem involves proving that if the sequence {An} converges to A, then the sequence {|An|} converges to |A|, and questioning whether the converse is true. This falls under the subject area of calculus, specifically dealing with sequences and limits.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss using the inequality ||x| - |y|| ≤ |x - y| as a potential approach to the proof. There are questions about how to apply this inequality within the formal epsilon definition of limits. Some participants suggest considering specific sequences, such as 1, -1, 1, -1, to explore the converse.

Discussion Status

The discussion is ongoing, with various participants offering insights and questioning the validity of certain approaches. There is no explicit consensus yet, but some guidance has been provided regarding the use of inequalities in the proof.

Contextual Notes

Participants are working under the constraints of needing to present their arguments in a formal proof format for a calculus class. There is some confusion regarding the application of the epsilon definition and the inequalities involved.

Mush89
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Homework Statement


Prove that if the sequence {An} converges to A, then the sequence {|An|} converges to |A|. And is the converse true?

This is for my calculus class and it needs to be in proof format. Thank you!


The Attempt at a Solution


I'm totally lost, I was going to use ||x| - |y|| less than/or equal to |x-Y| but I'm not really sure where to go from there and if that's even right.
 
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Just insert this inequality into the formal epsilon definition of the limit and you're done.
For the converse, consider the sequence 1,-1,1,-1,...
 
Last edited:
So, find the limit of that inequality?
 
What is the epsilon definition of the limit of a sequence?
 
Reversed Triangle Inequality will work:

l l An l - l A l l ≤ l An -A l*Use formal espilon definition
 
Last edited:
icystrike said:
l An - A l ≤ l An l - l A l
Above used for proving the Forward Direction

I don't think that works, because epsilon is on the right hand side ... you really need the other inequality already given by Mush89.
 
grey_earl said:
I don't think that works, because epsilon is on the right hand side ... you really need the other inequality already given by Mush89.

Opps! My bad! I guess the second inequality I've mentioned will be enough ..
 

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