Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be...

Recall that (a,b)⊆R means an open interval on the real number line:

(a,b)={x∈R|a<x<b}.

Let ≤ be the usual “less than or equal to” total order on the set

A=(−2,0)∪(0,2)

Consider the subset

B={−1/n | n∈N, n≥1}⊆A.

Determine an upper bound for B in A. Then formally prove that B has no least upper bound in A by arguing that every element of A fails the criteria in the definition of least upper bound.

Note:

least upper bound is an upper bound for B⊆A that is less than every other upper bound

Solutions

Expert Solution

Here is the required solution.I hope the answer will help you.Please give a thumbs up if you get benefited by my effort.Your feedback is very much precious to me.Thank you.

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Let R be the real line with the Euclidean topology. (a) Prove that R has a...

Let R be the real line with the Euclidean topology. (a) Prove that R has a countable base for its topology. (b) Prove that every open cover of R has a countable subcover.

Recall that the absolute value |x| for a real number x is defined as the function...

Recall that the absolute value |x| for a real number x is defined as the function |x| = x if x ≥ 0 or −x if x < 0. Prove that, for any real numbers x and y, we have (a) | − x| = |x|. (b) |xy| = |x||y|. (c) |x-1| = 1/|x|. Here we assume that x is not equal to 0. (d) −|x| ≤ x ≤ |x|.

f : [a, b] → R is continuous and in the open interval (a,b) differentiable. f...

f : [a, b] → R is continuous and in the open interval (a,b) differentiable. f rises strictly monotonously ⇒ ∀x ∈ (a, b) : f ′(x) > 0. (TRUE or FALSE?) f rises strictly monotonously ⇐ ∀x ∈ (a, b) : f ′(x) > 0. (TRUE or FALSE?) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?) If f is reversable, f has no critical point. (TRUE or FALSE?) If a is a “minimizer” of f, then f ′(a)...

15. Let r be a positive real number. The equation for a circle of radius r...

15. Let r be a positive real number. The equation for a circle of radius r whose center is the origin is (x^2)+(y^2)= r^2 . (a) Use implicit differentiation to determine dy/dx . (b) Let (a,b) be a point on the circle with a does not equal 0 and b does not equal 0. Determine the slope of the line tangent to the circle at the point (a,b). (c) Prove that the radius of the circle to the point (a,b)...

Let R be the region enclosed by the x-axis, the y-axis, the line x = 2...

Let R be the region enclosed by the x-axis, the y-axis, the line x = 2 , and the curve ? = 2?? +3? (1) Find the area of R by setting up and evaluating the integral. (2) Write, but do not evaluate, the volume of the solid generated by revolving R around the y-axis (3) Write, but do not evaluate the volume of the solid generated by revolving R around the x-axis (4) Write, but do not evaluate the...

Let L = {x = a r b s c t | r + s =...

Let L = {x = a r b s c t | r + s = t, r, s, t ≥ 0}. Give the simplest proof you can that L is not regular using the pumping lemma.

Let X have a uniform distribution on the interval [A, B]. (a) Obtain an expression for...

Let X have a uniform distribution on the interval [A, B]. (a) Obtain an expression for the (100p)th percentile, x. x = (b) Compute E(X), V(X), and σX. E(X) = V(X) = σX = (c) For n, a positive integer, compute E(Xn). E(Xn) =

Let R be the ring of all 2 x 2 real matrices. A. Assume that A...

Let R be the ring of all 2 x 2 real matrices. A. Assume that A is an element of R such that AB=BA for all B elements of R. Prove that A is a scalar multiple of the identity matrix. B. Prove that {0} and R are the only two ideals. Hint: Use the Matrices E11, E12, E21, E22.

Let x be a set and let R be a relation on x such x is...

Let x be a set and let R be a relation on x such x is simultaneously reflexive, symmetric, and antisymmetric. Prove equivalence relation.

(a) Let λ be a real number. Compute A − λI. (b) Find the eigenvalues of...

(a) Let λ be a real number. Compute A − λI. (b) Find the eigenvalues of A, that is, find the values of λ for which the matrix A − λI is not invertible. (Hint: There should be exactly 2. Label the larger one λ1 and the smaller λ2.) (c) Compute the matrices A − λ1I and A − λ2I. (d) Find the eigenspace associated with λ1, that is the set of all solutions v = v1 v2 to (A...

Subjects