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 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....

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.

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...

Let A be open and nonempty and f : A → R. Prove that f is
continuous at a if and only if f is both upper and lower
semicontinuous at a.

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.

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 ? ⊂ 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...

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