We should think of S as containing all the points OI S anu all puts as limits of convergent sequences of points of S. A slightly different formulation of th notion is given in Exercise 8. an be obtaine EXERCISES 2.2 1. Which of the following subsets of R" is open? closed? neither? Prove your answer. (a) x : 0x < 2} C R (b) {x x 2- for some keN or x= 0} C R (g) : y=xCR : X = y (h) x: 0 x |1}C R" (i) x x 1} C R" j x |x 1} C R" (k) the set of rational numbers, Q CR X (c) C R2 y X (d) y 2 у CR2 1 (1) X: ||X||<1 or X CR2 X (e) : у>х У C R2 (m) (the empty set) X (f) : ху 2 У CR2 2. Let {x} be a sequence of points in R". For i = 1, .. . , n, 1let xi denote the ith coordinate of the vector Xk. Prove that xk a if and only if xk,iai for all i = 1, .. . , n. 3. Suppose {x} is a sequence of points (vectors) in R" converging to a. (a) Prove that |Xk |l a . (Hint: See Exercise 1 17.)

We should think of S as containing all the points OI S anu all puts as limits of convergent sequences of points of S. A slightly different formulation of th notion is given in Exercise 8. an be obtaine EXERCISES 2.2 1. Which of the following subsets of R" is open? closed? neither? Prove your answer. (a) x : 0x < 2} C R (b) {x x 2- for some keN or x= 0} C R (g) : y=xCR : X = y (h) x: 0 x |1}C R" (i) x x 1} C R" j x |x 1} C R" (k) the set of rational numbers, Q CR X (c) C R2 y X (d) y 2 у CR2 1 (1) X: ||X||<1 or X CR2 X (e) : у>х У C R2 (m) (the empty set) X (f) : ху 2 У CR2 2. Let {x} be a sequence of points in R". For i = 1, .. . , n, 1let xi denote the ith coordinate of the vector Xk. Prove that xk a if and only if xk,iai for all i = 1, .. . , n. 3. Suppose {x} is a sequence of points (vectors) in R" converging to a. (a) Prove that |Xk |l a . (Hint: See Exercise 1 17.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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