Question

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

Answer #1

prove that a compact set is closed using the Heine - Borel
theorem

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...

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).

A function f is said to be Borel measurable provided its domain
E is a Borel set and for each c, the set {x in E l f(x) > c} is
a Borel set. Prove that if f and g are Borel measurable functions
that are defined on E and are finite almost everywhere on E, then
for any real numbers a and b, af+bg is measurable on E and fg is
measurable on E.

Analysis:
Show rigorously an example of a Lebesgue medible set, but not
borel.

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.

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).

Prove that the set of real numbers has the same cardinality
as:
(a) The set of positive real numbers.
(b) The set of nonnegative real numbers.

Prove that the set of real numbers has the same cardinality
as:
(a) The set of positive real numbers.
(b) The set of non-negative real numbers.

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.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 7 minutes ago

asked 16 minutes ago

asked 44 minutes ago

asked 56 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago