Question

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.

Answer #1

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).

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 S is a subset of the Reals and is non-empty and bounded
above. Prove that alpha = supS if and only if , for every epsilon
> 0 , there is and element x in S such that alpha - epsilon
<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

Using the completeness axiom, show that every nonempty set E of
real numbers that is bounded below has a greatest lower bound
(i.e., inf E exists and is a real number).

Let A be a nonempty set. Prove that the set S(A) = {f : A → A |
f is one-to-one and onto } is a group under the operation of
function composition.

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
⊂...

Let S be the set {(-1)^n +1 - (1/n): all n are natural
numbers}.
1. find the infimum and the supremum of S, and prove that these
are indeed the infimum and supremum.
2. find all the boundary points of the set S. Prove that each of
these numbers is a boundary point.
3. Is the set S closed? Compact? give reasons.
4. Complete the sentence: Any nonempty compact set has a....

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 S and T be nonempty subsets of R with the following
property: s ≤ t for all s ∈ S and t ∈ T.
(a) Show that S is bounded above and T is bounded below.
(b) Prove supS ≤ inf T .
(c) Given an example of such sets S and T where S ∩ T is
nonempty.
(d) Give an example of sets S and T where supS = infT and S ∩T
is the empty set....

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