By D. Drasin (auth.), Carlos A. Berenstein (eds.)
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Additional info for Complex Analysis III: Proceedings of the Special Year held at the University of Maryland, College Park, 1985–86
A) x0 D 34 , S D 5. (d) S D . 1; 1/ 1 ;1 2 (b) x0 D 23 , S D 1 3 ; 2 2 (c) x0 D 5, S D . 0; 2/ (a) S D . 1; 2/ [ Œ3; 1/ (b) S D . S 0 /c . S [ T /c D S c \ T c . Let F be a collection of sets and define ˚ ˇ « ˚ ˇ « I D \ F ˇF 2 F and U D [ F ˇ F 2 F : ˇ ˚ ˇ « ˚ « Prove that (a) I c D [ F c ˇ F 2 F and (b) U c D \F c ˇ F 2 F . (a) Show that the intersection of finitely many open sets is open. 28 Chapter 1 The Real Numbers (b) Give an example showing that the intersection of infinitely many open sets may fail to be open.
X/ D 1. x// D L. 26. x0 C 27. x/: x! x/ both exist in the extended reals and are equal, in which case all three are equal. In Exercises 28–30 consider only the case where at least one of L1 and L2 is ˙1. 28. x0 52 Chapter 2 Differential Calculus of Functions of One Variable 29. 1 30. x/ D cos x, and x0 D =2. 31. e x 32. 2 x 3 C 2x C 3 2x 4 C 3x 2 C 2 2x 4 C 3x 2 C 2 x! x/ and limx! x/ D where an ¤ 0 and bm ¤ 0. a0 C a1 x C b0 C b1 x C C an x n ; C bm x m 33. a; b/. a; b/? Justify your answer. 34.
16. (a) Prove: If S is bounded above and ˇ D sup S , then ˇ 2 @S . (b) State the analogous result for a set bounded below. 17. 18. 19. Prove: If S is closed and bounded, then inf S and sup S are both in S . 20. If a nonempty subset S of R is both open and closed, then S D R. Let S be an arbitrary set. Prove: (a) @S is closed. (b) S 0 is open. (c) The exterior of S is open. (d) The limit points of S form a closed set. (e) S D S . Give counterexamples to the following false statements. (a) (b) (c) (d) (e) (f ) (g) (h) The isolated points of a set form a closed set.