1. Let d : R x R → R be defined to bed(x, y) = | arctan(x) – arctan(y)|.Show that d is a metric on R.2. Show that (R, d), where d is the metric from Problem 1, is not a complete metric space. In otherwords, show that (R, d) has a Cauchy sequence that does not converge.

“Since you have posted multiple questions, we will provide the solution only to the first question…

Answered: 1. Let d: RxR → R be defined to be d(x,…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...

See similar textbooks

Similar questions

2. Let A {1,2,3} be equipped with the discrete metric d, (i.e. d(x, y) = 1 if x # y, and 0 otherwise), and let the Cartesian product E = [0,1] × A be equipped with the %3D d(x, y) do-metric (i.e. d»((x, a), (x', a')) = max{|r – x'|, d(a, a')}). Give examples of %3D %3D a) a compact subspace K CE, b) a non-compact subspace LC E, c) a connected subspace M CE, d) a disconnected subspace N C E, and justify your claims.
4. i) Let X = B[a, b] and let d be the sup metric on X. If In → x in (X,d) and an is continuous at to E [a, b] for each n > 1, prove that x is also continuous at to. %3D ii) Let In (t) convergent in B [0, 1] with the sup metric? t", for all t e [0, 1] and all integers n > 1. Is the sequence {¤n}
2. Prove that if d is a metric for X, then another metric d' for X is given by if a ≤ b for any real numbers " if b ≤ a (a) d'(x, y) a, b. (b) d'(x, y) = = min{1, d(x, y)}, where min{a, b} d(x,y) 1+d(x,y* = a b
2. Let (X, d) and (X, d') be a metric spaces such that there is K E N satisfying 1 d'(p, q) ≤ d (p, q) ≤ d' (p, q), (Vp, q € X) K Show that if (X, d') is a complete metric space, then (X, d) is also a complete metric space.
3.5 Show that the metric space (R, d), where d(x, y) = |x – y|, is separable. (4)
4. Let (X, d) be a metric space and d' be a metric on X that is equivalent to d. Let T and 7' be the topology on X induced by d and d' respectively. Show that I = I'.
Q2: (a) In Euclidean metric space (R, 1. ]),if A = {y € R: y = 2 cos (2x - 1). ; x = R} then fined the following A,A, À. (b) In Euclidean metric space (R², |. ||),if A = {(x, y): x² + y² <1} determine if the set A is bounded or not bounded set in R². 11:52 لعد و دقاق
5. Let X R2 with metric d((x1, y₁), (2, y2)) = √(x₂-x₁)² + (y2 - y₁)². Prove that the sequences 2) converges. Is it a Cauchy sequences?
2 Q2:- Let (X, d) be a metric space. Xo, Yo EX and r, s > 0. Show that B, (yo) ≤ Br(xo) if s+d(xo, yo) ≤r.

Recommended textbooks for you

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

Calculus For The Life Sciences

Calculus

ISBN:

9780321964038

Author:

GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:

Pearson Addison Wesley,

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

Calculus For The Life Sciences

Calculus

ISBN:

9780321964038

Author:

GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:

Pearson Addison Wesley,

SEE MORE TEXTBOOKS

1. Let d: RxR → R be defined to be d(x, y) = |arctan(x) - arctan(y)]. Show that d is a metric on R. 2. Show that (R, d), where d is the metric from Problem 1, is not a complete metric space. In other words, show that (R, d) has a Cauchy sequence that does not converge.