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 RXCR2y=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 RXr(c)X-(d)1X-10(1I)X: |x1 orC R2(e)CR2(m) Ø (the empty set)X(f)xy 0 R2y2. Let {x&} be a sequence of points in R". For i = 1, .. . , n, let xki denote the ith coordinate of thevector x. Prove that xk a if and only if xkiai for all i = 1, ....n3. 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.

Question
Asked Aug 9, 2019
1 views

I need help for the proof of question 2. Thank you very much!

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.
help_outline

Image Transcriptionclose

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.

fullscreen
check_circle

Expert Answer

star
star
star
star
star
1 Rating
Step 1

Given: 

help_outline

Image Transcriptionclose

A sequence of points {x^} in R". For i=1,...,n, let x, denote the th coordinate of the vector x

fullscreen
Step 2

To prove:

help_outline

Image Transcriptionclose

if and only if x>a, for all i=1..,n 1

fullscreen
Step 3

Proof:

...
help_outline

Image Transcriptionclose

Consider R" to be a Euclidean space with 1-norm, then x|=^x. i-1 Let us assume x > a

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*

See Solution
*Response times may vary by subject and question.
Tagged in

Math

Advanced Math

Related Advanced Math Q&A

Find answers to questions asked by student like you
Show more Q&A
add