26. Let a = (a₁,a2,a3,...,an) be an n-tuple of positiv ■d (w₁,w2,W3,..., Wn, ‚wn) be another n-tuple of positive real num =). Define the weighted mean of order r by 1/r Σ;=1 w;a; if r 9. and Irl<

9.K Let X = Q y K = {p € Q : 2 < p? < 3} . Show that K is closed in Q, is bounded, and is not compact in Q.

The Real Numbers *7. Let S be a nonempty bounded subset of R and let ke R. Define kS = {ks :se S}. Prove the following: (a) If k20, then sup (kS) = k · sup S and inf (kS) =k · inf S. (b) Ifk 0, then there exists ne N such that n-1< y < n. ☆ (b) Prove that the n in part (a) is unique. 10. (a) Prove: If x and y are real numbers with x < y, then there are infinitely many rational numbers in the interval [x, y]. (b) Repeat part (a) for irrational numbers. 11. Let y be a positive real number. Prove that for every n e N there exists a I unique positive real number x such that x" = y. ☆ *12. Let D be a nonempty set and suppose that f: D → R and g: D → R. Define the function f +g:D→R by (f+g)(x)= f(x) +g(x). (a) If f(D) and g(D) are bounded above, then prove that (f + g)(D) is bounded above and sup[(ƒ +g)(D)] < sup f(D) + sup g(D). (b) Find an example to show that a strict inequality in part (a) may occur. (c) State and prove the analog of part (a) for infima. 13. Let x e R. Prove that x = sup…

5. Does f(x) = In(x) satisfy the hypotheses of the Mean Value Theorem on [1,4]? If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem.