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

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.

sweet thanks