In: Advanced Math

Prove that if a sequence is bounded, then it must have a convergent subsequence.

please give a sequence in the form an=
that is monotonic, bounded and convergent.

1. prove that if{xn} is decreasing an bounded from
below, then {xn} is convergent.

Prove that if a sequence is bounded, then limsup sn is a real
number.

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...

Let (an) be a real sequence in the standard metric. Prove that
(an) is bounded if and only if every subsequence of (an) has a
convergent subsequence.

Let (Xn) be a monotone sequence. Suppose that (Xn) has a Cauchy
subsequence. Prove that (Xn) converges.

Show that every cauchy sequence of reals is convergent

Prove that if f is a bounded function on a bounded interval
[a,b] and f is continuous except at finitely many points in [a,b],
then f is integrable on [a,b]. Hint: Use interval additivity, and
an induction argument on the number of discontinuities.

Prove Longest common subsequence algorithm class finds the
optimal solution

Let sequence an be a bounded sequence and let E be the
set of subsequential limits of an. prove that E is bounded and
contains both sup E and inf E

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