Is the Converse of the Sequence Converging Proof True?

[Mush89]

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.

 

[grey_earl]

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,...

 

[Mush89]

So, find the limit of that inequality?

 

[grey_earl]

What is the epsilon definition of the limit of a sequence?

 

[icystrike]

Reversed Triangle Inequality will work:

l l An l - l A l l ≤ l An -A l*Use formal espilon definition

 

[grey_earl]

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.

 

[icystrike]

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 ..