Convergence question: example where |a_n| -> 0 but a_n does not

[Steven G]

Can someone please give me an example of a sequence a_n where |a_n| converges to 0 as n-->oo but a_n does not? Explaining how to come up with these sequences would be greatly appreciated. Intuitively it seems that a_n does not exist but .....
Thanks!

 

[stapel]

Can someone please give me an example of a sequence a_n where |a_n| converges to 0 as n-->oo but a_n does not? Explaining how to come up with these sequences would be greatly appreciated. Intuitively it seems that a_n does not exist but .....

On what basis had you concluded that such a sequence existed? (In comparison, see Theorem 2 here or Theorem 11.1.4 here.)

 

[Steven G]

Sorry, I wrote the statement wrong. Of course what I stated is true. I apologize for the time people spent on my original post.

I meant to say if someone can prove or give a counter example to: If a_n converges to 0 as n-->oo, then |a_n| converges to 0 as n-->oo.

I actually came up with a proof. Can someone please verify it? It is attached.

Thanks,
Jomo

 

[pka]

Can someone please give me an example of a sequence a_n where |a_n| converges to 0 as n-->oo but a_n does not? Explaining how to come up with these sequences would be greatly appreciated. Intuitively it seems that a_n does not exist but .....

TRY $\displaystyle \large a_n=[(-1)^n+n^{-1}]$.

 

[Steven G]

TRY $\displaystyle \large a_n=[(-1)^n+n^{-1}]$.

If I am not missing something I get that a_n does not converge and |a_n| converges to 1. Neither a_n nor |a_n| converges to 0