Sequence Converging Proof

  • Thread starter Mush89
  • Start date
  • #1
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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.
 

Answers and Replies

  • #2
169
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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,...
 
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  • #3
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So, find the limit of that inequality?
 
  • #4
169
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What is the epsilon definition of the limit of a sequence?
 
  • #5
446
1
Reversed Triangle Inequality will work:

l l An l - l A l l ≤ l An -A l


*Use formal espilon definition
 
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  • #6
169
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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.
 
  • #7
446
1
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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