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.