Question

Suppose A ⊆ R is nonempty and bounded above and β ∈ R. Let A + β = {a + β : a ∈ A}

Prove that A + β has a supremum and sup(A + β) = sup(A) + β.

Answer #1

Let A⊆R be a nonempty set, which is bounded above. Let
B={a-5:a∈ A}. Prove that sup(B)=sup(A)-5

let A be a nonempty subset of R that is bounded below. Prove
that inf A = -sup{-a: a in A}

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.

Question 3. Let a1,...,an ∈R. Prove that
(a1 + a2 + ... + an)2
/n ≤ (a1)2 + (a2)2 +
... + (an)2.
Question 5. Let S ⊆R and T ⊆R be non-empty. Suppose that s ≤ t for
all s ∈ S and t ∈ T. Prove that sup(S) ≤ inf(T).
Question 6. Let S ⊆ R and T ⊆ R. Suppose that S is bounded above
and T is bounded below. Let U = {t−s|t ∈ T, s...

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.

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...

Let f : [a,b] → R be a bounded function and let:
M = sup f(x)
m = inf f(x)
M* =sup |f(x)|
m* =inf |f(x)|
assuming you are taking values of x that lie in
[a,b].
Is it true that M* - m* ≤ M - m ?
If it is true, prove it. If it is false, find a counter
example.

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

1. Let A ⊆ R and p ∈ R. We say that A is bounded away from p if
there is some c ∈ R+ such that |x − p| ≥ c for all x ∈ A. Prove
that A is bounded away from p if and only if p not equal to A and
the set n { 1 / |x−p| : x ∈ A} is bounded.
2. (a) Let n ∈ natural number(N) , and suppose that k...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 10 minutes ago

asked 15 minutes ago

asked 19 minutes ago

asked 34 minutes ago

asked 47 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 3 hours ago

asked 3 hours ago