Math 74 Final Exam Practice ProblemsDecember 9, 2008Easier Problems1. Show that Z is a closed subset of R in the Euclidean metric.2. Let (X, d) be a metric space and let Y ⊆ X be any subset. SinceY × Y ⊆ X × X, the function d : X × X → R restricts to a functiondY: Y × Y → R. Show that (Y, dY) is a metric space.3. For each of the three definitions of continuity, give a proof that theidentity function 1X: X → X on any metric space (X, d) is a contin-uous function.4. Using the definition of convergence, show that the sequence (sn) in Rdefined bysn=nXi=112nconverges to 1.5. (Not a problem, a “friendly suggestion.”) Review the construction ofZ, Q, and R.6. Let (X, d) and (Y, ρ) be metric spaces, and let f : X → Y be afunction. Use quantifier negation and the epsilon-delta definition togive a careful definition of what it means for f to not be continuous.Use this to show that the function f : R → R,f(x) =0 x ≤ 01 x > 0is not continuous.17. Let a, b ∈ Z, and suppose that ab ≡ 2(mod 4). Show that either a orb is even, but not both.Medium Problems8. Use the epsilon-delta definition of continuity to show that the functionf : R → R defined by f(x) = x3is continuous. Give another (easier)proof of this fact using the techniques developed in class.9. Let A ⊆ R be a closed subset (in the Euclidean metric) and let r ∈ Rbe arbitrary. Show that the set A + r := {a + r | a ∈ A} is closed.10. Let (X, d) and (Y, ρ) be metric spaces, let f : X → Y be a continuousfunction, and let A ⊆ X be an arbitrary subset. Show that f (¯A) ⊆f(A) (where the bars here stand for closure). Give an example thatshows that f(¯A) can be a proper subset of f (A).11. Prove the squeeze theorem: if (xn), (yn), and (zn) are sequences in Rsuch that:(a) Both (xn) and (zn) converge, and limnxn= limnzn, and(b) For all n ∈ N \ {0}, we have xn≤ yn≤ zn,then the sequence (yn) also converges, and limnyn= limnxn.12. Let X and Y be sets, let f : X → Y be a function, and let A be asubset of Y . Show that f (f−1(A)) = f(X) ∩ A.13. Let a ∈ N \ {0} be arbitrary, and leta = pr11· pr22· · · prnnbe the prime factorization of a. Find a formula in terms of the riforthe number of distinct natural numbers which divide a. Prove thatyour answer is correct.14. Prove the binomial theorem: if a, b ∈ R and n ∈ N, then(a + b)n=nXi=0niaibn−i.15. Show that if p is prime and 1 ≤ i ≤ p − 1, thenpiis divisible by p.Give an example that shows that this can be false if p is not prime.2Harder Problems16. Let (X, d) be a metric space. Suppose that for each n ∈ N \ {0},Yn⊆ X is a subset of X which is Cauchy complete (with the metricrestricted from X). MustSn∈NYnbe Cauchy complete? What aboutTn∈NYn?17. Show that R is a connected metric space. [Outline: start by assumingthat U ⊆ R is both open and closed, and that U 6= R and U 6= ∅. LetV = R \ U , and let x ∈ U be arbitrary. Show that for some r ∈ R,r > 0, either (x, x + r) ∩ V or (x − r, x) ∩ V is nonempty. Assuming(x, x + r) ∩ V is nonempty, let s be the greatest lower bound on theset {r ∈ R | r > 0, (x, x + r) ∩ V 6= ∅}. Show that x + s is an elementof both U and V .]18. Let a ∈ R be the least upper bound for the set {q ∈ Q | q2≤ 2} ⊆ R.Explain why a exists, and show that a2=