Question

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.

Answer #1

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...

Prove: Let x and y be bounded sequences such that xn
≤ yn for all n ∈ N. Then lim supn→∞
xn ≤ lim supn→∞ yn and lim
infn→∞ xn ≤ lim infn→∞
yn.

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 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

For a given real number x , there is a natural number n which is
larger than x .
True
False
The supremum of the set of negative integers is 0.
True
False
The supremum of a bounded set of rational numbers is
rational.
True
False
The supremum of a bounded set of irrational numbers is
irrational.
True
False
Every rational number is the supremum of a bounded set of
irrational numbers.
True
False
Every bounded sequence is a Cauchy...

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).

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.

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
⊂...

Let x and y be real numbers. Then prove that sqrt(x^2) = abs(x)
and abs(xy) = abs(x) * abs(y)

(a) Let f(z) = z^2. R is bounded by y = x, y= -x and x = 1. Find
the image of R under the mapping f.
(b)Find the all values of (-i)^(i)
(c)Find the all values of (1-i)^(4i)

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