Question

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.

Answer #1

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 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.

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...

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....

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).

Let S be a nonempty set in Rn, and its support
function be
σS = sup{ <x,z> : z ∈ S}.
let conv(S) denote the convex hull of S. Show that σS
(x)= σconv(S) (x), for all x ∈ Rn

Suppose K is a nonempty compact subset of a metric space X and
x∈X.
Show, there is a nearest point p∈K to x; that is, there
is a point p∈K such that, for all other q∈K,
d(p,x)≤d(q,x).
[Suggestion: As a start, let S={d(x,y):y∈K} and show there is a
sequence (qn) from K such that the numerical sequence (d(x,qn))
converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}.
Show, there is a point z∈X and distinct points a,b∈T
that are nearest points to...

Let R be a ring.
Show that R[x] is a finitely generated R[x]-module if and only if
R={0}.
Show that Q is not a finitely generated Z-module.

Let (X , X) be a measurable space. Show that f : X → R is
measurable if
and only if {x ∈ X : f(x) > r} is measurable for every r ∈
Q.

. Let M be an R-module; if me M let 1(m) = {x € R | xm = 0}.
Show that 1(m) is a left-ideal of R. It is called the order of m.
17. If 2 is a left-ideal of R and if M is an R-module, show that
for me M, λm {xm | * € 1} is a submodule of M.

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