Show that a point x is an accumulation point of a set E if and only...

Show that a point x is an accumulation point of a set E if and only if for every ε > 0 there are at least two points belonging to the set E ∩ (x − ε, x + ε).

Expert Solution

Colby Messinger answered 2 years ago

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is...

(a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset of R^k...

ONLY NEED PART E PLEASE SHOW ALL WORK Company X projects numbers of unit sales for...

ONLY NEED PART E PLEASE SHOW ALL WORK Company X projects numbers of unit sales for a new project as follows: 81,000 (year 1), 89,000 (year 2), 97,000 (year 3), 92,000 (year 4), and 77,000 (year 5). The project will require $1,500,000 in net working capital to start (year 0) and require net working capital investments each year equal to 15% of the projected sales for subsequent years (year 1 - 5). NWC is recovered at the end of the...

Show that the set of rigid motions E(3) forms a group.

let S in Rn be convex set. show that u point in S is extreme point...

let S in Rn be convex set. show that u point in S is extreme point of S if and only if u is not convex combination of other points of S.

Show that a set is convex if and only if its intersection with any line is...

Show that a set is convex if and only if its intersection with any line is convex. Show that a set is affine if and only if its intersection with any line is affine.

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that...

(Connected Spaces) (a) Let <X, d> be a metric space and E ⊆ X. Show that E is connected iff for all p, q ∈ E, there is a connected A ⊆ E with p, q ∈ E. b) Prove that every line segment between two points in R^k is connected, that is Ep,q = {tp + (1 − t)q | t ∈ [0, 1]} for any p not equal to q in R^k. C). Prove that every convex subset...

A photon with an E nerge is scattered by making an x angle. Show that the...

A photon with an E nerge is scattered by making an x angle. Show that the scattered photon energy is E '= E / (1+ (2E / mc ^ 2) sin ^ 2 (x / 2)).

Let X and Y be random variables with finite means. Show that min g(x) E(Y−g(X))^2=E(Y−E(Y|X))^2 Hint:...

Let X and Y be random variables with finite means. Show that min g(x) E(Y−g(X))^2=E(Y−E(Y|X))^2 Hint: a−b = a−c+c−b

Prove Corollary 4.22: A set of real numbers E is closed and bounded if and only...

Prove Corollary 4.22: A set of real numbers E is closed and bounded if and only if every infinite subset of E has a point of accumulation that belongs to E. Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real numbers is closed and bounded if and only if every sequence of points chosen from the set has a subsequence that converges to a point that belongs to E. Must use Theorem 4.21 to prove Corollary 4.22 and there should...

Show that a set S has infinite elements if and only if it has a subset...

Show that a set S has infinite elements if and only if it has a subset U such that (1) U does not equal to S and (2) U and S have the same cardinality.

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