2. Let A be a non-empty subset of R bounded below. Show that inf (A) is...

Question

Question

2. Let A be a non-empty subset of R bounded below. Show that inf (A) is
a border point of A

Answer 1

Answer #1

Solution :- Cet a be any non emply subset of IR which is bounded below. > InHCA) escisty Suppose int(A) = a Now to show a is

Answer 2

Similar Homework Help Questions

Let A be a non empty subset of R that is bounded below and let a=inf...

Let A be a non empty subset of R that is bounded below and let a=inf A. If a a&A, prove thal x is a limit point of A
5. Let S be a non empty bounded subset of R. If a > 0, show...

5. Let S be a non empty bounded subset of R. If a > 0, show that sup (as) = a sup S where as = {as : ES}. Let c = sup S, show ac = sup (aS). This is done by showing (a) ac is an upper bound of aS. (b) If y is another upper bound of as then ac S7 Both are done using definitions and the fact that c=sup S.
Let A be a non-empty subset of R that is bounded above. (a) Let U =...

Let A be a non-empty subset of R that is bounded above. (a) Let U = {x ∈ R : x is an upper bound for A}, the set of all upper bounds for A. Prove that there exists a u ∈ R such that U = [u, ∞). (b) Prove that for all ε > 0 there exists an x ∈ A such that u − ε < x ≤ u. This u is one shown to exist in...

#3 A Supremely Infimum Problem (Zorn 1.9 #8) Let S R be non-empty and bounded below. Let-S f-xlxES). Show that sup(-S) exists. Then show that -inf (S) sup(-S). This problem shows that the completenes...

#3 A Supremely Infimum Problem (Zorn 1.9 #8) Let S R be non-empty and bounded below. Let-S f-xlxES). Show that sup(-S) exists. Then show that -inf (S) sup(-S). This problem shows that the completeness axiom guaranteeing the existence of supremums implies a similar statement about the existence of infimums. Write down an "infimum" version of the completeness axiom. that-1 #3 A Supremely Infimum Problem (Zorn 1.9 #8) Let S R be non-empty and bounded below. Let-S f-xlxES). Show that sup(-S)...
No Contradiction 2. Let A and B be non-empty subsets of R, and suppose that ACB....

No Contradiction 2. Let A and B be non-empty subsets of R, and suppose that ACB. Prove that if B is bounded below then inf B <inf A.
5. Let S be a non-empty bounded subset of R. If a > 0, show that...

5. Let S be a non-empty bounded subset of R. If a > 0, show that sup (aS) = a sup S where aS = {as : s E S}. Let c = sup S, show ac = sup (aS). This is done by showing: (a) ac is an upper bound of aS. (b) If y is another upper bound of aS then ac < 7. Both are done using definitions and the fact that c=sup S.

Use the completeness axiom to show that every non empty subset of R (real numbers) that...

Use the completeness axiom to show that every non empty subset of R (real numbers) that is bounded below has an infimum in R
6. Let X be a non-empty subset of an ordered field with the least upper bound...

6. Let X be a non-empty subset of an ordered field with the least upper bound property. Supposed that X is bounded above and define -X = {-1 : TEX} Prove that supX = - inf(-X).
(6) Let S c R be non-empty and bounded above. Let q = sup S. Show that q E bd S.

(6) Let S c R be non-empty and bounded above. Let q = sup S. Show that q E bd S. (6) Let S c R be non-empty and bounded above. Let q = sup S. Show that q E bd S.

Let A be a nonempty subset of R. Define -A={-a: a A}. (a) Prove that if...

Let A be a nonempty subset of R. Define -A={-a: a A}. (a) Prove that if A is bounded below, then -A is bounded above. (b) Prove that if A is bounded below, then A has an infimum in R and inf A=-sup (-A).