Let 1 ≤ p < ∞ and let {fn} be a sequence of functions in Lp[a, b]. Suppose there is a function f ∈ Lp[a, b] with limn→∞ ||fn − f||p = 0 . Prove that {fn} is a Cauchy sequence in Lp[a, b].

1. Construct a continuous function f : X → Y between two metric spaces with the condition that there exists a Cauchy sequence {n}₁ in X but {f(xn)}~_=₁ is not a Cauchy sequence in Y. n=1

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).

Give an example of a bounded function that is defined on R, continuous at each point except the points 1 and-1, and has neither a removable discontinuity nor a jump discontinuity at 1 and-1.