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.)

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I need help for problem 1c. Thanks!

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.)
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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.)