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.
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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I need help for the proof of question 2. Thank you very much!
Transcribed Image Text: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.
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