\text { Problem 5: If }\left\{s_{n}\right\} \text { is a complex sequence, define its arithmetic means } \sigma_{n} \text { by } \sigma_{n}=\frac{s_{0}+s_{1}+\cdots+s_{n}}{n+1} \quad(n=0,1,2, \ldots) \text { 1. If } \lim s_{n}=s \text {, prove that } \lim \sigma_{n}=s \text {. } \text { 2. Construct a sequence }\left\{s_{n}\right\} \text { which does not converge although } \lim \sigma_{n}=0 \text {. }

Fig: 1

Fig: 2

Fig: 3

Fig: 4