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.
Then you need to write out a formal proof for yourself.
Here is what it will be based upon, $\displaystyle \left||x|-|y|\right|\le |x-y|$.
I was going to plug that inequality into the definition of a limit, but is that going in the wrong direction?
It absolutely does not.
The sequence $\displaystyle (a_n)$ converges. That is the one we have control over. We use it the get a handle on the sequence $\displaystyle \left|(a_n)\right|$