Question

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

Answer #1

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).

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).

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...

Show that a bounded decreasing sequence converges to its
greatest lower bound.

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.

Find the least upper bound and the greatest lower bound for the
two polynomials:
a) p(x) = x4 - 3x2 - 2x + 5
b) p(x) = -2x5 + 5x4 + x3 - 3x
+ 4

For n in natural number, let A_n be the subset of all those real
numbers that are roots of some polynomial of degree n with rational
coefficients.
Prove: for every n in natural number, A_n is countable.

Real Topology: let A={1/n : n is natural} be a subset of the real
numbers. Is A open closed, or neither? Justify your answer.

why
is every countable subset a zero set? real analysis

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...

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