EXERCISES 2.2 "1. Which of the following subsets of R" is open? closed? neither? Prove your answer. (a) (x :0 x< 2} CR (b) (x:x= 2 for some k eN or =0) C R X CR2 y=x (g) y (h) {x: 0 |x|| < 1} C R" (i) x |x 1} C R" (G) x ||xl1} C R" (k) the set of rational numbers, Q C R Xr (c) X- (d) 1 X- 10 (1I) X: |x 1 or C R2 (e) CR2 (m) Ø (the empty set) X (f) xy 0 R2 y 2. Let {x&} be a sequence of points in R". For i = 1, .. . , n, let xki denote the ith coordinate of the vector x. Prove that xk a if and only if xki ai for all i = 1, ....n 3. Suppose (x} is a sequence of points (vectors) in R" converging to a. (a) Prove that |x all. (Hint: See Exercise 1.2.17.) (b) Prove that if b e R" is any vector, then b xk b a.

EXERCISES 2.2 "1. Which of the following subsets of R" is open? closed? neither? Prove your answer. (a) (x :0 x< 2} CR (b) (x:x= 2 for some k eN or =0) C R X CR2 y=x (g) y (h) {x: 0 |x|| < 1} C R" (i) x |x 1} C R" (G) x ||xl1} C R" (k) the set of rational numbers, Q C R Xr (c) X- (d) 1 X- 10 (1I) X: |x 1 or C R2 (e) CR2 (m) Ø (the empty set) X (f) xy 0 R2 y 2. Let {x&} be a sequence of points in R". For i = 1, .. . , n, let xki denote the ith coordinate of the vector x. Prove that xk a if and only if xki ai for all i = 1, ....n 3. Suppose (x} is a sequence of points (vectors) in R" converging to a. (a) Prove that |x all. (Hint: See Exercise 1.2.17.) (b) Prove that if b e R" is any vector, then b xk b a.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 63RE

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