Question

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).

Answer #1

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.

Let S be the set of real numbers between 0 and 1, inclusive;
i.e. S = [0, 1]. Let T be the set of real numbers between 1 and 3
inclusive (i.e. T = [1, 3]). Show that S and T have the same
cardinality.

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).

Prove Corollary 4.22: A set of real numbers E is closed and
bounded if and only if every infinite subset of E has a point of
accumulation that belongs to E.
Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real
numbers is closed and bounded if and only if every sequence of
points chosen from the set has a subsequence that converges to a
point that belongs to E.
Must use Theorem 4.21 to prove Corollary 4.22 and there should...

Let S be the collection of all sequences of real numbers and
define a relation on S by {xn} ∼ {yn} if and only if {xn − yn}
converges to 0.
a) Prove that ∼ is an equivalence relation on S.
b) What happens if ∼ is defined by {xn} ∼ {yn} if and only if
{xn + yn} converges to 0?

Let R*= R\ {0} be the set of nonzero real
numbers. Let
G= {2x2 matrix: row 1(a b) row 2 (0 a) | a in
R*, b in R}
(a) Prove that G is a subgroup of GL(2,R)
(b) Prove that G is Abelian

Define the 'closure' S of a set S of real numbers.
State as many equivalent characterizations, or results, about
the closure S.

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

Let (sn) ⊂ (0, +∞) be a sequence of real numbers. Prove that
liminf 1/Sn = 1 / limsup Sn

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

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