Question

Problem 2 Let {xn) , (yn) and {zn^ be three number sequences and suppose that for some fixed integer K, we have Please check whether following statements are true or false. If true, prove them. If false, provide at least one counter-example. (i) The convergence of rn implies the convergence of yn (ii) The convergence of zn implies the convergence of yn. (ii) The convergence of n, zn to some number L implies the convergence of yn (iv) The divergence of yn implies the divergence of zn (v) The divergence of xn to infinity implies the divergence of zn (vi) The convergence of Σ000zn implies the convergence of Σ_oyn. (vii) The convergence of Ση00 zn implies the convergence of Ση00xn (viii) The divergence of Ση.oYn implies the divergence of Ση- Zn (ix) The convergence of Σ00oyn implies the convergence of Σ; zn (x) The divergence of Ση-0zn implies the divergence of Σ00ozn

Answer 1