Question

Let S and T be nonempty subsets of R with the following property: s ≤ t...

Let S and T be nonempty subsets of R with the following property: s ≤ t for all s ∈ S and t ∈ T.

  1. (a) Show that S is bounded above and T is bounded below.

  2. (b) Prove supS ≤ inf T .

  3. (c) Given an example of such sets S and T where S ∩ T is nonempty.

  4. (d) Give an example of sets S and T where supS = infT and S ∩T is the empty set. In your explanation, make sure you justify why sup S = inf T .

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
let A be a nonempty subset of R that is bounded below. Prove that inf A...
let A be a nonempty subset of R that is bounded below. Prove that inf A = -sup{-a: a in A}
Question 3. Let a1,...,an ∈R. Prove that (a1 + a2 + ... + an)2 /n ≤...
Question 3. Let a1,...,an ∈R. Prove that (a1 + a2 + ... + an)2 /n ≤ (a1)2 + (a2)2 + ... + (an)2. Question 5. Let S ⊆R and T ⊆R be non-empty. Suppose that s ≤ t for all s ∈ S and t ∈ T. Prove that sup(S) ≤ inf(T). Question 6. Let S ⊆ R and T ⊆ R. Suppose that S is bounded above and T is bounded below. Let U = {t−s|t ∈ T, s...
Suppose S and T are nonempty sets of real numbers such that for each x ∈...
Suppose S and T are nonempty sets of real numbers such that for each x ∈ s and y ∈ T we have x<y. a) Prove that sup S and int T exist b) Let M = sup S and N= inf T. Prove that M<=N
Let A⊆R be a nonempty set, which is bounded above. Let B={a-5:a∈ A}. Prove that sup(B)=sup(A)-5
Let A⊆R be a nonempty set, which is bounded above. Let B={a-5:a∈ A}. Prove that sup(B)=sup(A)-5
Suppose A ⊆ R is nonempty and bounded above and β ∈ R. Let A +...
Suppose A ⊆ R is nonempty and bounded above and β ∈ R. Let A + β = {a + β : a ∈ A} Prove that A + β has a supremum and sup(A + β) = sup(A) + β.
Is there a set A ⊆ R with the following property? In each case give an...
Is there a set A ⊆ R with the following property? In each case give an example, or a rigorous proof that it does not exist. d) Every real number is both a lower and an upper bound for A. (e) A is non-empty and 2inf(A) < a < 1 sup(A) for every a ∈ A.2 (f) A is non-empty and (inf(A),sup(A)) ⊆ [a+ 1,b− 1] for some a,b ∈ A and n > 1000.
3. Assume √2 ∈R. Let S = { rational numbers q : q < √2 }....
3. Assume √2 ∈R. Let S = { rational numbers q : q < √2 }. (a)(i) Show that S is nonempty. (ii) Prove that S is bounded from above, but is not bounded from below. (b) Prove that supS = √2.
Let f : [a,b] → R be a bounded function and let:             M = sup...
Let f : [a,b] → R be a bounded function and let:             M = sup f(x)             m = inf f(x)             M* =sup |f(x)|             m* =inf |f(x)| assuming you are taking values of x that lie in [a,b]. Is it true that M* - m* ≤ M - m ? If it is true, prove it. If it is false, find a counter example.
1. (a) Let S be a nonempty set of real numbers that is bounded above. Prove...
1. (a) Let S be a nonempty set of real numbers that is bounded above. Prove that if u and v are both least upper bounds of S, then u = v. (b) Let a > 0 be a real number. Define S := {1 − a n : n ∈ N}. Prove that if epsilon > 0, then there is an element x ∈ S such that x > 1−epsilon.
Prove: Let S be a bounded set of real numbers and let a > 0. Define...
Prove: Let S be a bounded set of real numbers and let a > 0. Define aS = {as : s ∈ S}. Show that inf(aS) = a*inf(S).
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT