answer all parts please! :)Homework Answers
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answer all parts please! :) 1. Recall that (an) which is positively (resp textitnegatively) bounded away...
1. Recall that x E R is positive (resp. negative) if x = (an) which is...
1. Recall that x E R is positive (resp. negative) if x = (an) which is positively (resp textitnegatively) bounded away from 0. Prove the following LIM00 an for a Cauchy sequence n-oo (a) For any E R, exactly one of the following is true: x is positive, is negative, or x= 0 E R is positive if and only if -x is negative. (b) (c) If x, y E R are both positive, then x + y and xy...
problem 23 please :) and here is Q.21 Problem 23. Recall from Problem 21 the equivalence...
problem 23 please :)
and here is Q.21
Problem 23. Recall from Problem 21 the equivalence relation ~ on the set of rational Cauchy sequences C. Define 〈z) E C to be eventually positive if there is an M є N such that xn > 0 for all Prove that eventually positive is a well defined notion on c/ (z〉 ~ 〈y), then 〈y〉 İs eventually positive. ie. if 〈z) is eventually positive and Problem 21. Let C be the...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
Correction: first problem is #2, not #1. Please show all steps in the proofs. Definitions for...
Correction: first problem is #2, not #1. Please show all steps
in the proofs.
Definitions for problems #2 through #5: Let C be the set of all Cauchy sequences of rational numbers, with the operations of addition and multiplication defined on C by (an) + (bn) = (an + bn) and (an)(bn) = (anbn). Let N be the subset of C consisting of all null sequences in c. Properties of a ring: A1. (a + b) +c= a + b...
With justification in each one. Clarification; why if true and why if false? Please Determine whether...
With justification in each one. Clarification; why if true and why if false? Please Determine whether the following statement is true or false: • Iff: R+R is differentiable and strictly increasing on R, then f'(1) > 0 VI ER • If S: R R is continuous and f(x) - ron Q, then (V3) - 3. • If f,g: (0,1) - Rare functions such that \S(1)-f(y) = g(1)-9(y) for all 1, y € (0, 1) and g is continuous on (0,1),...
(10 points.) Recall that a real number a is said to be rational if a =...
(10 points.) Recall that a real number a is said to be rational if a = " for some m,n e Z and n +0. (a) Use this definition to prove that if and y are both rational numbers, then r+y is also rational (b) Prove that if r is rational and y is irrational, then x+y is irrational
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....
e=[(0,1)] Recall that a rational number x = [(a,b)] is positive if ab > 0 in...
e=[(0,1)]
Recall that a rational number x = [(a,b)] is positive if ab > 0 in Z and negative if ab < 0 in Z (for any choice of representative (a,b) E x). For x,y EQ we say x <y iff y e(-x) is positive. (a) Show that x <y iff x (-y) is negative. (b) Show that for each x E Q, precisely one of the following three statements is true: x = e, e < x, x<e. (c)...
Solve all parts please 5. In the following problems, recall that the adjacency matrix (or incidence matrix) for a simp...
Solve all parts please
5. In the following problems, recall that the adjacency matrix (or incidence matrix) for a simple graph with n vertices is an n x n matrix with entries that are all 0 or 1. The entries on the diagonal are all 0, and the entry in the ih row and jth column is 1 if there is an edge between vertex i and vertex j and is 0 if there is not an edge between vertex...
Please answer in the style of a formal proof and thoroughly reference any theorems, lemmas or...
Please answer in the style of a formal proof and thoroughly
reference any theorems, lemmas or corollaries utilized.
BUC
stands for bounded uniformly continuous
Let (X, d) be a metric space. Show that the set V of Lipschitz continu- ous bounded functions from X to R is a dense linear subspace of BUC(X, R). Since, in general, V #BUC(X, R), V is not a closed subset of BUC(X, R). Hint: For f EBUC(X, R) define the sequence (fr) by fn(x)...
1. Recall that x E R is positive (resp. negative) if x = (an) which is positively (resp textitnegatively) bounded away from 0. Prove the following LIM00 an for a Cauchy sequence n-oo (a) For any E R, exactly one of the following is true: x is positive, is negative, or x= 0 E R is positive if and only if -x is negative. (b) (c) If x, y E R are both positive, then x + y and xy...
problem 23 please :)
and here is Q.21
Problem 23. Recall from Problem 21 the equivalence relation ~ on the set of rational Cauchy sequences C. Define 〈z) E C to be eventually positive if there is an M є N such that xn > 0 for all Prove that eventually positive is a well defined notion on c/ (z〉 ~ 〈y), then 〈y〉 İs eventually positive. ie. if 〈z) is eventually positive and Problem 21. Let C be the...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
Correction: first problem is #2, not #1. Please show all steps
in the proofs.
Definitions for problems #2 through #5: Let C be the set of all Cauchy sequences of rational numbers, with the operations of addition and multiplication defined on C by (an) + (bn) = (an + bn) and (an)(bn) = (anbn). Let N be the subset of C consisting of all null sequences in c. Properties of a ring: A1. (a + b) +c= a + b...
With justification in each one. Clarification; why if true and why if false? Please Determine whether the following statement is true or false: • Iff: R+R is differentiable and strictly increasing on R, then f'(1) > 0 VI ER • If S: R R is continuous and f(x) - ron Q, then (V3) - 3. • If f,g: (0,1) - Rare functions such that \S(1)-f(y) = g(1)-9(y) for all 1, y € (0, 1) and g is continuous on (0,1),...
(10 points.) Recall that a real number a is said to be rational if a = " for some m,n e Z and n +0. (a) Use this definition to prove that if and y are both rational numbers, then r+y is also rational (b) Prove that if r is rational and y is irrational, then x+y is irrational
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....
e=[(0,1)]
Recall that a rational number x = [(a,b)] is positive if ab > 0 in Z and negative if ab < 0 in Z (for any choice of representative (a,b) E x). For x,y EQ we say x <y iff y e(-x) is positive. (a) Show that x <y iff x (-y) is negative. (b) Show that for each x E Q, precisely one of the following three statements is true: x = e, e < x, x<e. (c)...
Solve all parts please
5. In the following problems, recall that the adjacency matrix (or incidence matrix) for a simple graph with n vertices is an n x n matrix with entries that are all 0 or 1. The entries on the diagonal are all 0, and the entry in the ih row and jth column is 1 if there is an edge between vertex i and vertex j and is 0 if there is not an edge between vertex...
Please answer in the style of a formal proof and thoroughly
reference any theorems, lemmas or corollaries utilized.
BUC
stands for bounded uniformly continuous
Let (X, d) be a metric space. Show that the set V of Lipschitz continu- ous bounded functions from X to R is a dense linear subspace of BUC(X, R). Since, in general, V #BUC(X, R), V is not a closed subset of BUC(X, R). Hint: For f EBUC(X, R) define the sequence (fr) by fn(x)...
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