Question

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

Answer #1

Suppose A and B are
nonempty sets of real numbers, and that for every x
∈ A, and every y ∈ B, we have x < y. Prove that A ≤
inf(B).

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....

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.

For each of the following sets S, find sup(S) and inf(S) if they
exist:
(a) {x ∈ R| x2 < 100}
(b) {-3/n | n is a counting number}
(c) {.9, .99, .999, .9999, .99999, …}

Suppose A is a subset of R (real numbers) sucks that both infA
and supA exists. Define -A={-a: a in A}.
Prive that:
A. inf(-A) and sup(-A) exist
B. inf(-A)= -supA and sup(-A)= -infA
NOTE:
supA=u defined by: (u is least upper bound of A) for all x in A,
x <= u, AND if u' is an upper bound of A, then u <= u'
infA=v defined by: (v is greatest lower bound of A) for all y in...

Let
x = {x} and y ={y} represent bounded sequences of real numbers, z =
x + y, prove the following: supX + supY = supZ where sup represents
the supremum of each sequence.

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
⊂...

A. Let p and r be
real numbers, with p < r. Using the axioms of
the real number system, prove there exists a real number q
so that p < q < r.
B. Let f: R→R be a polynomial
function of even degree and let A={f(x)|x
∈R} be the range of f. Define f
such that it has at least two terms.
1. Using the properties and
definitions of the real number system, and in particular the
definition...

Suppose S ⊂ R is nonempty and M is an upper bound for S. Show M
= sup S if and
only if for every Ɛ > 0, there exists x ∈ S so that x > M −
Ɛ.

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).

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