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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!

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- Thread starter MathSquareRoo
- Start date

In this case, you are using a definition of convergence to prove a statement about convergence.So, in summary, the statement is proved using a definition of convergence.

- #1

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Prove that the sequence {a_n} converges to A if and only if lim n--->∞ (a_n-A)=0.

It's an if and only if proof, but I'm not sure how to prove it. Please help!

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- #2

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Try writing each statement in epsilon-delta form, and compare.

- #3

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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?

- #4

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MathSquareRoo said:

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?

Well, how did you prove it in the forward direction? Can you simply reverse the reasoning?

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I don't know if this is correct, and I don't know where to go after that.

- #6

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MathSquareRoo said:

I don't know if this is correct, and I don't know where to go after that.

Yes, that's correct. But what's the difference between this:

([itex]\lim_{n \rightarrow \infty} (a_n - A) = 0)[/itex]: "So given epsilon>0, there exists N>0 s.t. la_n-Al<epsilon for all n>N."

versus what you wrote earlier:

([itex]\lim_{n \rightarrow \infty} a_n = A)[/itex]: "Given epsilon>0, there exists N>0 s.t. lan-Al<epsilon for all n>N."

- #7

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I'm not sure. I don't know if I have it written correctly. I feel like I'm working in circles.

- #8

Science Advisor

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Applying exactly the same definition, "[itex](a_n- A)[/itex] converges to 0"

But [itex](a_n- A)- 0[/itex]

Remember that definitions in mathematics are "working definitions"- you use the precise words of definitions in problems and proofs.

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