Question

Real Analysis I Prove the following exercises (show all your work)- Exercise 1.1.1: Prove part (iii)...

Real Analysis I

Prove the following exercises (show all your work)-

Exercise 1.1.1: Prove part (iii) of Proposition 1.1.8. That is, let F be an ordered field and x, y,z ∈ F. Prove If x < 0 and y < z, then xy > xz.

Let F be an ordered field and x, y,z,w ∈ F. Then:

If x < 0 and y < z, then xy > xz.

Exercise 1.1.5: Let S be an ordered set. Let A ⊂ S and suppose b is an upper bound for A. Suppose b ∈ A. Show that b = sup A.

Exercise 1.1.6: Let S be an ordered set. Let A ⊂ S be a nonempty subset that is bounded above. Suppose sup A exists and sup A ∈/ A. Show that A contains a countably infinite subset.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Using field and order axioms prove the following theorems: (i) Let x, y, and z be...
Using field and order axioms prove the following theorems: (i) Let x, y, and z be elements of R, the a. If 0 < x, and y < z, then xy < xz b. If x < 0 and y < z, then xz < xy (ii) If x, y are elements of R and 0 < x < y, then 0 < y ^ -1 < x ^ -1 (iii) If x,y are elements of R and x <...
Suppose K is a nonempty compact subset of a metric space X and x∈X. Show, there...
Suppose K is a nonempty compact subset of a metric space X and x∈X. Show, there is a nearest point p∈K to x; that is, there is a point p∈K such that, for all other q∈K, d(p,x)≤d(q,x). [Suggestion: As a start, let S={d(x,y):y∈K} and show there is a sequence (qn) from K such that the numerical sequence (d(x,qn)) converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}. Show, there is a point z∈X and distinct points a,b∈T that are nearest points to...
Calculus III. Please show all work and mark the answer(s)! 1) Use Lagrange multipliers to find...
Calculus III. Please show all work and mark the answer(s)! 1) Use Lagrange multipliers to find the maximum and minimum values of the function f(x, y) = x^2 + y^2 subject to the constraint xy = 1. 2) Use Lagrange Multipliers to find the point on the curve 2x + 3y = 6 that is closest to the origin. Hint: let f(x, y) be the distance squared from the origin to the point (x, y), then find the minimum of...
5. Suppose that the incenter I of ABC is on the triangle’s Euler line. Show that...
5. Suppose that the incenter I of ABC is on the triangle’s Euler line. Show that the triangle is isosceles. 6. Suppose that three circles of equal radius pass through a common point P, and denote by A, B, and C the three other points where some two of these circles cross. Show that the unique circle passing through A, B, and C has the same radius as the original three circles. 7. Suppose A, B, and C are distinct...