![7. Denoting, as usual, by (a, b) an open interval, {x : a < x < b} and by [a,b] the corresponding closed interval, {x : a < x < b} of real numbers, prove that supla, b) = sup a, b = b and inf(a, b) = inf[a,b- other words, does the proof work for any ordered field? a. Is completeness relevant:In](http://img.homeworklib.com/questions/9623abc0-82ad-11eb-a098-efdb848f85d0.png?x-oss-process=image/resize,w_560)
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7. Denoting, as usual, by (a, b) an "open interval", {x : a < x <...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if C>0, then 7 is also integrable on la,b] (6 Marks)...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if C>0, then 7 is also integrable on la,b] (6 Marks) (2) If C 0, i, still integrable (assuming f(x)关0 for any x E [aM)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval [a, b]....
ANSWER 5,6 & 7 please. Show work for my understanding and upvote. THANK YOU!! Problem 5....
ANSWER 5,6 & 7 please. Show work for my understanding and
upvote. THANK YOU!!
Problem 5. (3 pts) Let {x,n} be a bounded sequence of real numbers and let E = {xn : n E N}. Prove that lim inf,,0 In and lim inf, Yn are both in E. Hint: Use the sequential characterization of the closure, i.e., Proposition 3.2 from class. Problem 6. (3 pts) As usual let Q denote the set of all rational numbers. Prove that R....
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if С > 0, then, is also integrable on [a,b, (6 Marks)...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if С > 0, then, is also integrable on [a,b, (6 Marks) (2) If C 0, i, still integrable (assuming f(x) 0 for any x E [aA)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval...
(b) Prove that R is numerically equivalent to any bounded open or closed interval.
(b) Prove that R is numerically equivalent to any bounded open or closed interval.
Let g: R→R be a polynomial function of even degree and let B={g(x) ER) be the...
Let g: R→R be a polynomial function of even degree and let B={g(x) ER) be the range of g. Define g such that it has at least two terms. 1. Using the properties and definitions of the real number system, and in particular the definition of infimum, construct a formal proof showing inf(B) exists OR explain why B does not have an infimum. 2. Using the properties and definitions of the real number system, and in particular the definition of...
Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be...
Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be the usual “less than or equal to” total order on the set A=(−2,0)∪(0,2). Consider the subset B={−1/n | n∈N,n≥1}⊆A. Determine an upper bound for B in A.. Then formally prove that B has no least upper bound in A by arguing that every element of A fails the criteria in the definition of least upper bound. Note: make sure you are addressing the technical...
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2....
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
Supposef is continuous on the closed interval [a, b]. if F(x) is an antiderivative of f(x)...
Supposef is continuous on the closed interval [a, b]. if F(x) is an antiderivative of f(x) for all real numbers a,b and k, then the definite integral S* (R8) + k]dx is equal to (A F(b) - F(a) + b-a B F(b) - F(a) +k(b-a) C F(b + k) - F(a + k)+k k F(b)-k F(a) + b-a
Suppose we tried to apply our real analysis definitions/methods to the set of rational numbers Q....
Suppose we tried to apply our real analysis definitions/methods
to the
set of rational numbers Q. In other words, in the definitions, we
only
consider rational numbers. E.g., [0, 1] now means [0, 1] ∩ Q, etc.
In
this setting:
(a) Find an open cover of [0, 1] that contains no finite subcover.
Hint:
Fix an irrational number α ∈ [0, 1] (as a subset of the reals
now!)
and for each (rational) q ∈ [0, 1] look for an...
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if C>0, then 7 is also integrable on la,b] (6 Marks) (2) If C 0, i, still integrable (assuming f(x)关0 for any x E [aM)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval [a, b]....
ANSWER 5,6 & 7 please. Show work for my understanding and
upvote. THANK YOU!!
Problem 5. (3 pts) Let {x,n} be a bounded sequence of real numbers and let E = {xn : n E N}. Prove that lim inf,,0 In and lim inf, Yn are both in E. Hint: Use the sequential characterization of the closure, i.e., Proposition 3.2 from class. Problem 6. (3 pts) As usual let Q denote the set of all rational numbers. Prove that R....
12. Let f be integrable on a closed interval [a, b]. Suppose that there is a real number C such that f(x) 2C for all E a, b (1) Prove that if С > 0, then, is also integrable on [a,b, (6 Marks) (2) If C 0, i, still integrable (assuming f(x) 0 for any x E [aA)? If yes, supply a short proof. If no, give a counterexample. (6 Marks)
12. Let f be integrable on a closed interval...
(b) Prove that R is numerically equivalent to any bounded open or closed interval.
Let g: R→R be a polynomial function of even degree and let B={g(x) ER) be the range of g. Define g such that it has at least two terms. 1. Using the properties and definitions of the real number system, and in particular the definition of infimum, construct a formal proof showing inf(B) exists OR explain why B does not have an infimum. 2. Using the properties and definitions of the real number system, and in particular the definition of...
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
Supposef is continuous on the closed interval [a, b]. if F(x) is an antiderivative of f(x) for all real numbers a,b and k, then the definite integral S* (R8) + k]dx is equal to (A F(b) - F(a) + b-a B F(b) - F(a) +k(b-a) C F(b + k) - F(a + k)+k k F(b)-k F(a) + b-a
Suppose we tried to apply our real analysis definitions/methods
to the
set of rational numbers Q. In other words, in the definitions, we
only
consider rational numbers. E.g., [0, 1] now means [0, 1] ∩ Q, etc.
In
this setting:
(a) Find an open cover of [0, 1] that contains no finite subcover.
Hint:
Fix an irrational number α ∈ [0, 1] (as a subset of the reals
now!)
and for each (rational) q ∈ [0, 1] look for an...
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