Question

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 A, v <= y, AND if v' is a lower bound of A, then v' <= v

Answer #1

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 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

Show that every nonempty subset of the real numbers with a lower
bound has a greatest lower bound.

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.

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).

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
⊂...

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).

Suppose ? ⊂ R^? , ? ⊂ R^? are nonempty and open and ? : ? → R^?
and ? : ? → R^? . Let ℎ : ? × ? → R ?+? be defined by ℎ(u, v) =
(?(u), ?(v)). If ? is continuous at x ∈ ? and ? is continuous at y
∈ ? , then show that ℎ is continuous at (x, y) ∈ ? × ? .
Hint: Note that for any vectors z...

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.

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.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 8 minutes ago

asked 17 minutes ago

asked 39 minutes ago

asked 48 minutes ago

asked 55 minutes ago

asked 1 hour ago

asked 2 hours ago

asked 3 hours ago

asked 3 hours ago

asked 3 hours ago

asked 3 hours ago

asked 3 hours ago