Sequences and Series, Terms Approach Zero

Terms Approach Zero

If s is a convergent series that approaches p, let t be the sequence of partial sums. For any ε, find n such that all terms beyond t_n are within ε/2 of p. Thus the difference t_j-t_j-1, for j beyond n, is less than ε. This difference is s_j, hence the terms of s approach 0.

If the members of s come from an additive metric space, such as the complex numbers, |s_j| approaches 0, hence s_j approaches the origin.