Question

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).

Answer #1

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).

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

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.

Suppose A is an mxn matrix of real numbers, suppose
x is in the null space of A and suppose
y is in the column space of AT. prove
that x is orthogonal to y

Let A and B be nonempty sets. Prove that if f is an injection,
then f(A − B) = f(A) − f(B)

Let A and B be nonempty sets. Prove that if f is an injection,
then f(A − B) = f(A) − f(B)

10. Prove if X and Y are nonempty closed subsets of [a,b]⊂ ℝ
such that X∪Y=[a,b], then X∩Y≠ ∅.

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 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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