Proof of that a limit -> 0 as n-> infinity

Hey,

I've been trying to work out the following question,

This is it including what I hope is an ok, or on the way to an ok proof.

http://img269.imageshack.us/img269/6156/proofxq.jpg

The book has a hint given that states there exists N in naturals such that |z_n|<ε/2 for all n>N so I kind of tried to use that to find a bound on w_n.

Does this look okay?

Thank you in advance,

Jess

Re: Proof of that a limit -> 0 as n-> infinity

It is not true that for every ε there exists an N such that |z_{N+1}+ ... +zₙ| < ε / 2 for all n > N. E.g., for zₙ = 1 / n, any tail z_{N+1} + z_{N+2} + ... is infinite.

It is even less true that |z₁ + ... + z_{N}| < ε / 2. For example, we could have z₁ = 100.

Re: Proof of that a limit -> 0 as n-> infinity