Question

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).

Answer #1

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

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).

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 Corollary 4.22: A set of real numbers E is closed and
bounded if and only if every infinite subset of E has a point of
accumulation that belongs to E.
Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real
numbers is closed and bounded if and only if every sequence of
points chosen from the set has a subsequence that converges to a
point that belongs to E.
Must use Theorem 4.21 to prove Corollary 4.22 and there should...

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).

Show that if a set of real numbers E has the Heine-Borel
property then it is closed and bounded.

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. 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 (an) is an increasing sequence of real numbers. Show, if
(an) has a bounded subsequence, then (an) converges; and (an)
diverges to infinity if and only if (an) has an unbounded
subsequence.

14. Show that if a set E has positive outer measure, then there
is a bounded subset of E that also
has positive outer measure.

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