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...
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.