2. Give examples (without proofs) of the following: a) sequences {xn} and {yn} which both diverge, but wh {Xn+yn} converges.

Suppose that the sequence {n} is generated by the fixed point iteration method xn = g(xn-1), n ≥ 1. Further, suppose that the sequence convergence linearly to the fixed point p. Show that (a) g'(p) ~ _In-n-1 Xn-1 Xn-2 9 g' (p) (b) |en| ~₁ |In — Xn-1|- g' (p)-1

Q(A). Let {fn(x)}1 = be a sequence of functions 1+ (x – 2)" defined over [2,3]. Show that: (a) fn(x) is meaurable and monotonic increasing for all n. (b) {fn(x)}-1 converges a.e. to a function f(x) to be determined. (c) Apply the monotone convergence theorem to evaluate Lim fn(x)dµ. (B) Let {fn(x)}1 = {x ln x. cos(x – 1)"}1 be a sequence of meaurable functions defined over [1,2]. Show that: (a) {fn(x)}1 converges to a function f(x) to be determined. (b) Show that fn(x) is dominated by some integrable function for all n. Then apply the dominated convergence theorem to evaluate Lim x In x.cos(x- 1)"dµ.