Which of the following statements are correct?
a) If A is a bounded subset of the real line, every infinite
subset of A has a limit point.
b) If A is a bounded subset of the real line, every open cover
of A has a finite subcover.
c) If A is an infinite open subset of the real line, there is an
infinite open cover with a finite subcover.
d) If A is a closed subset of the real line,...
Let {an} be a bounded sequence. In this question,
you will prove that there exists a convergent subsequence.
Define a crest of the sequence to be a
term am that is greater than all subsequent terms. That is,
am > an for all n > m
(a) Suppose {an} has infinitely many crests. Prove
that the crests form a convergent subsequence.
(b) Suppose {an} has only finitely many crests. Let
an1 be a term with no subsequent crests. Construct a...
Suppose that a subset S of an ordered field F
is not bounded above in F. Let T be a subset of
F satisfying the property that, for each x ∈
S, there exists y ∈ T such that
x ≤ y. Prove that T is not bounded above
in F.
Show that if Y is a subspace of X, and A is a subset of Y, then
the subspace topology on A as a subspace of Y is the same as the
subspace topology on A as a subspace of X.
Let G be an abelian group and K is a subset of G.
if K is a subgroup of G , show that G is finitely generated if
and only if both K and G/K are finitely generated.