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using the sandwich theorem there. but the converse i cant figure out

What is an in this context? a function?

a sequence

like an = (-1)^n / n^2 for n = 1,2,3...

So basically, prove:
$$\lim a_n = 0 \iff \lim |a_n| = 0$$
Correct?

How do you define a limit?

well the <= direction i can prove using sandwich theorem

for each epsilon there exists an N such that if n>N then
|an - L | < e

you mean we have to prove | |an| - 0 | < e

so there are two cases

now we will show that for e* > 0 there exists an N* such that if n>N* then | |an| - 0 | < e*

ok so far?

cant you do it directly , taking the case when an>0 and an < 0

Yea, I think this is can be done with a simple direct proof as you said.

nomo, your proof looks good

davis, theres a big wrinkle in the direct proof

well, what if an is both positive and negative

no what you wrote is correct

the proof looks like a direct consequence of the definition of the limit

but your proof works as well