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Homework 5 Assignment: Read pages 83 to 90 in Chapter 4, but skip Theorem 4.10. Examples 4.11, Definition 4.18, Theorem 4.19, Theorem 4.20. Pay attention to my proof of the Intermediate Value Theorem in class! Read the section “Infinite Limits and Limits at Infinity” (bottom of page 97 to page 98). Do these problems: 1. Prove, straight from the definition in Section 4.1, that lim x2 + x = 6. x→2 (You need to tell me what δ to use for any given ϵ.) 2. Show that lim sin(1/x) x→0 does not exist. 3. Suppose X is a metric space, f and g are functions from X to R, and f is bounded (that is, the image f (X) is bounded in R). Suppose also that for some p ∈ X, lim g(x) = 0. x→p Show that lim f (x)g(x) = 0. x→p 4. Suppose X is a metric space and f is a continuous real-valued function on X. Show that the set {x ∈ X : f (x) ≤ 0} is closed. 5. Suppose f and g are continuous functions from R to R. (a) Suppose f (r) = 0 for every rational number r. Show that f (x) = 0 for all real x. (b) Suppose f (r) = g(r) for every rational number r. Must it be true that f (x) = g(x) for all real x? 6. Suppose f is a function from X to X, where X is a complete metric space, and that for some fixed constant c < 1 we have dX (f (x), f (y)) ≤ cdX (x, y) for all x, y ∈ X. 1 (a) Show that f is continuous. (b) Suppose we choose x1 ∈ X and define a sequence via xk+1 = f (xk ) for k ≥ 1. Show that this sequence is Cauchy. Explain why this sequence has a limit. (c) Let the limit of the sequence in the last part be denoted by x∗ . Show that x∗ = f (x∗ ). (d) Show that x∗ is the ONLY point in X that satisfies the equation x = f (x). 7. Decide on the validity of the following assertions (with proof/counterexamples, of course!) (a) Continuous functions on R map bounded open sets to bounded open sets. (b) Continuous functions on R map bounded open sets to open sets. (c) Continuous functions on R map bounded closed sets to bounded closed sets. (d) Continuous functions on R map bounded closed sets to closed sets. 8. Prove that the polynomial p(x) = x7 − 3x4 + x + 8 has at least one real root. 2