3. Assume that f : R → R is such that |f(x) – f(y)| < A\x – y| for all x, y E R and some A E (0, 1). Pick xo E R arbitrarily, and construct a sequence (xn) as follows: xn+1 f(xn) for n > 0. (a) Prove that |æn+1 – Xn] < \|xn – xn-1| for all n > 1. (b) Prove that |æn+1– Xn[ < \"|x1 – xo] for all n > 1. (c) Prove that (xn) is convergent. (d) Let x* lim xn. Prove that f(x*) = x*. ||

Let f be a real-valued function with domain R defined by f(x) = x2.  Define a sequence (s1, s2, s3, ...) by s1 = b and sj = f(sj-1), for j = 1,2,3,4...   a) assume that b=1, and find a formula for sn. b) assume that b=3, and find a formula for sn. c) prove the conjecture in part (b).

Consider the function f : R → R, and define the sequence {an} such that an = f(n). True or False? If lim n→∞ an = L, then lim x→∞ f(x) = L If true prove it, if false,provide a counterexample (i.e. a specific function for which the statement does not hold).

Consider the function f : R → R, and define the sequence {an} such that an = f(n). True or False? If limn→∞ an = l, then limx→∞ f(x) = l. If true prove it, if false, provide a counterexample (i.e. a specific function for which the statement does not hold).