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Part 1 Additional Problem 1: Let x = (x1, . . . , xn) ∈ R n . Show that Xn k=1 xk !2 ≤ n|x| 2 . Give necessary and sucient conditions on the numbers xk which guarantee equality in this inequality. Additional Problem 2: Let r > 0 and S = {x ∈ R 3 : x 2 1 + x 2 2 + x 2 3 < r2}. Show that diam S = 2r. Additional Problem 3: Show that the set E = {x ∈ R 2 : x1, x2 ∈ Q}. is dense in R 2 . Part 2 Additional Problem 1: Show that the intersection of sets from any family of closed sets is closed. Additional Problem 2: Show that any closed set F ⊂ R n is of type Gδ. Then use that result to show that any open set Ee ⊂ R n is of type Fσ. Part 3 2. Find lim sup Ek and lim inf Ek if Ek = [−(1/k), 1] for k odd and Ek = [−1,(1/k)] for k even. 3. a) Show that C(lim sup Ek) = lim inf CEk. 7. Show that E˚1 ∩ E˚2 = (E1 ∩ E2) ◦ , and E˚1 ∪ E˚2 ⊂ (E1 ∪ E2) ◦ . Give an example when E˚1 ∪ E˚2 6= (E1 ∪ E2) ◦ . 11. Give an example of decreasing sequence of nonempty closed sets in R n whose intersection is empty. 12. Give an example of two disjoint closed sets F1 and F2 in R n for which dist(F1, F2) = 0.

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