Prove, by definition, that if then .

It should be a basic proof but I'm somehow stuck. I was obviously thinking:

but I'm not sure how to proceed.

Any help would be appreciated.

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- Sep 19th 2012, 05:39 AMloui1410Proof - limit of a sequence
Prove, by definition, that if then .

It should be a basic proof but I'm somehow stuck. I was obviously thinking:

but I'm not sure how to proceed.

Any help would be appreciated. - Sep 19th 2012, 05:44 AMemakarovRe: Proof - limit of a sequence
is small and is close to 2L and therefore cannot be large.

- Sep 19th 2012, 05:51 AMloui1410Re: Proof - limit of a sequence
I don't understand...

- Sep 19th 2012, 06:00 AMemakarovRe: Proof - limit of a sequence
You are given an ε > 0 and you need to make |aₙ - L|⋅|aₙ + L| < ε by choosing n. You can make |aₙ - L| arbitrarily small by choosing n big enough. Also, you can make |aₙ + L| arbitrarily close to 2L by choosing n big enough. In particular, you can make |aₙ + L| < 3L for some n and onward. Assuming you have done this, how small do you need to make |aₙ - L| so that |aₙ - L|⋅|aₙ + L| becomes smaller than ε?