I need help with finding the proof and logic of sequence (ii). It is similar to that of sequence (i) provided bellow:

2.2.2 Continuity of algebraic operations Limits interact nicely with algebraic operations. Proposition 2.2.5. Let {xn} and {yn} be convergent sequences. (i) The sequence { Zn}, where zn := In + In, converges and lim (Xn + yn) = lim zn = lim Xn + lim yn. n-too n-too n-too n-too (ii) The sequence {Zn}, where Zn := Xn -yn, converges and lim (Xn - yn) = lim zn = lim Xn - lim yn. n-too n-too n-too n-ooProof. Let us start with (i). Suppose {xn} and {yn} are convergent sequences and write zn : = Xn + yn. Let x := lim Xn, y : = lim yn, and z := xty. Let & > 0 be given. Find an Mi such that for all n 2 Mi we have |Xn - x| M2 we have lyn - yl M we have | Zn - z) = 1(xn + yn) - (xty)| = [xn- xtyn-yl