{1- neN}. Find an open cover O = subcover. Prove that O is an open cover and that O has no finite subcover. Let E n+1 {On | n e N} of E that has no finite

Q: Let A be a nonempty subset of R. If a = sup A is finite, show that for each ɛ > 0, there is an a E A…

A: We have to prove the given statement

Q: Theorem 6.2. Let C be a compact subset of Rstd· Then C has a maximum point, that is, there is a…

A: We start the proof by assuming there are no such maximum element in C.

Q: Z[V2] denote the set {a + bV2|a, bɛZ}-Show that Z[V2] * Let is a subring of R. [See Example 20.]

A:

Q: Show that D^2 = {(x, y) ∈ E^2: x^2+y^2 ≤ 1} ⊂ E^2 and the space containing a single point are…

A: In algebraic topology, two topological spaces are said to be homotopy equivalent if there exists a…

Q: F, there is a disconnection A, B of F such that x E A and y E B. 8.X. If C1, C2 are connected…

A: We use the fact that a set X is connected if there is no continuous function form f from X to Y=…

Q: or which compact sets A does there exist an m e N such that every open cover of A contains a bcover…

A: Please check the answer in next step

Q: Draw a sketch of the following subsets of R² ([−3,3] × [−3, 3]) − ([−1, 1] × [−1,1]). - UKEN ([1, 2]…

A:

Q: The class of all finite disjoint unions of intervals of the form (a, b), (-0o, b), (b, o0), 0, and R…

A: Given: The class of all finite disjoint union of intervals of the form (a,b] ,(-∞,b] ,(b ,∞) ,∅ and…

Q: 4. Let S be a bounded subset of R and let So be a nonempty subset of S. Prove that inf Sinf So sup…

A: Let's prove the given inequalities : inf S ≤ inf S0:By definition of infimum, the infimum of a set S…

Q: Theorem 6.2. Let C be a compact subset of Rstd- Then C has a maximum point, that is, there is a…

A:

Q: Let S be a non empty subset of R and let u e R be such that u + is an upper bound of S for each n e…

A:

Q: Let D be an n-vertex digraph with no cycles. Prove that V (D) can be ordered as v1, . . . , vn such…

A:

Q: Suppose that A is a nonempty set of real numbers, and suppose that a ER is an upper bound for A.…

A:

Q: If the given three vectors are coplanar, then find the value of x. A = i-2j + 3k, B = xj + 3k, C =…

A: A, B, C are coplanar.So

Q: 2|A:k||2|| Bx:||2) < || |||B||F k%3D1

A: Please check the answer in next step

Q: In the set R of real numbers, define C, = (n,n+1) = {xeR|n < x <n+1} for each integer n. Determine…

A:

Q: Let i, r, and k be positive integers, and let s gcd(i, r). Assume that r | ik. Prove that | k.

A: To prove the divisibility property under the given conditions