Question 1 : If d, and d, are metrics on the same set X and there are positive integers a and b such that for all x, y E X, ad, (x, y) < d>(x, y) < bd1 (x, y). Show that the Cauchy sequences in (X,d1) and (X, d2) are the same.

Q: 1. (a) Consider the set S {n(-1)" : n e N}. Verify the existence or otherwise of Min(S) and Max(S).…

A: given that (1) (a)                                    S=n(-1)n:n∈N=2k(-1)2k:n=2k,…

Q: 4. Consider the set of functions S = {-3(x² – 4)°,7|x² – 4|³}. a. Show that S is a linearly…

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Q: (e) For the set W = {[1], [3], [13]} ≤ Z18 determine the set C = h¯¹(W). Then for the set C C Z12…

A: Modulo function

Q: 6. Let F = (-6, 5)U{-10, 10} U [7, 11). Find max F, min F, sup F, and inf F if they exist.

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Q: Let S = { (0,0), (0,3), (1,0), (1,2), (2,0), (3,2) } Find S', the transitive closure of S

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Q: Let A(0, π) υ (π, e 1| Find the limit point(s) of A. а. b. Is A an open set?

A: Given set is A = 0,π ∪ (π,e2]

Q: If A is bounded below and B is bounded above, show that inf(A − B) = inf A − sup B

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Q: (#4) Show that a set SCR" is closed if and only if bdry(S) CS.

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Q: Let A, BCR x R defined by A= {{n,m)|n, m € N, 0 < 2m(n + 1) – 3 < 5}, and B = {(1,1), (1,2), (2, 1),…

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Q: Let 13 20 x 20+ y = 2 be the equation of a curve in the xy-plane. Find the equation y = T(x) of the…

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Q: a. Write o using the two row form. b. Write o using cycle notation. c. Write the function o¬1 in the…

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Q: Theorem 2.22. Let A and B be subsets of a topological space X. Then: (1) A C B implies Ā c B. (2)…

A: Theorem 2.2 (i) Let A and B subsets of a topological space X such that A⊂B i.e. A is contained in B.…

Q: U is a neighborhood of x0 and U ⊂ V, The V is a neighborhood of x0. (2) If U1 ...., U2 are…

A: Solution ::

Q: Find all limit points of a set B = {1, 1 2 , 1 3 , 1 4 , . . .} on the real line with the usual…

A: The set B is B=1, 12, 13, 14, . . . . To Do: We have to find the limit point of set B on the real…

Q: Let S = {1,2,3, 4}. For each of the following relations on S give its transitive closure. a) {(1,1),…

A: Concept: In mathematics, the transitive closure of a binary relation R on set X is the smallest…

Q: In R4, let S = span{(1, 1, 1, 1)T ,(1, 2, 1, 3)T }. Find S⊥.

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Q: Proposition 2.3. Let X,Y be (topological) sets containing E and F, re- spectively. Moreover, let &…

A: Let X and Y be sets containing E and F respectively.  Suppose E and F be elementary families of X…

Q: 2. Let U = {a + bx + cx2 + d³ e P3 (R) | a- 6+d3D0}, and let W = {a + bx + ca? + dx³ e P3(R)| a + 2c…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.