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Homework Statement
Prove that the sequence {a_n} converges to A if and only if lim n--->∞ (a_n-A)=0.
Homework Equations
The Attempt at a Solution
It's an if and only if proof, but I'm not sure how to prove it. Please help!
Well, how did you prove it in the forward direction? Can you simply reverse the reasoning?I'm not good at writing proofs. So far I have:
Let {an} converge to A. Given epsilon>0, there exists N>0 s.t. lan-Al<epsilon for all n>N.
So l((a_n)-A)l<epsilon for all n>n.
Thus, we can write lim n--->infinity (a_n-A)=0.
Then, I'm not sure how to prove the statement's converse. Can someone help?
Yes, that's correct. But what's the difference between this:I'm not sure how to write the reverse. Would I start with: lim n--->infinity(a_n -A)=0. So given epsilon>0, there exists N>0 s.t. la_n-Al<epsilon for all n>N.
I don't know if this is correct, and I don't know where to go after that.