- #1

- 26

- 0

## 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!

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter MathSquareRoo
- Start date

- #1

- 26

- 0

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!

- #2

- 3,472

- 251

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

- #3

- 26

- 0

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

- 3,472

- 251

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?

- #5

- 26

- 0

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

- #6

- 3,472

- 251

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

- 26

- 0

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

- #8

HallsofIvy

Science Advisor

Homework Helper

- 41,833

- 963

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.

Share: