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10 (
a
) is a subsequence of (
a
).
(i) (a) Is it possibleto have
n
n
? (b) Is it possibleto have
n
n
? (c) What is theleast possiblevalueof
n
for any
subsequence? (d) Name a term of (
a
) which you know comes
later in the sequence (
a
) than
a
.
(ii) If
i
j
, does it follow that
n
n
and that
a
a
?
11 A sequence (
a
) is known to bemonotonic increasing, but not
strictly monotonic increasing.
(i) Might there be a strictly monotonic increasing subsequence of
(
a
)?
(ii) Must there be a strictly monotonic increasing subsequence of
(
a
)?
12 If a sequence is bounded, must each of its subsequences be
bounded?
13
(i) If the subsequence
a
,
a
,
a
, ...,
a
, . . . is bounded, does it
follow that the sequence (
a
) is bounded?
(ii) If the subsequence
a
,
a
,
a
, ...,
a
, . . . is bounded, does it
follow that the sequence (
a
) is bounded?
(iii) If the subsequence
a
,
a
,
a
, ...,
a
, . . . is bounded,
does it follow that the sequence (
a
) is bounded?
We describe the result of qn 13 by saying that if a sequence is
eventually bounded
, then it is
bounded
. Each of the sequences of qn 13 is
called a
shift
of the sequence (
a
). Obviously every shift of a bounded
sequence is bounded (qn 12), and from qn 13, if a shift is bounded, then
the whole sequence is bounded.
14
(a) If the subsequence (
a
) is bounded, does it follow that (
a
)is
bounded?
(b) A sequence (
a
) is known to beunbounded.
(i) Might it contain a bounded subsequence?
(ii) Must it contain a bounded subsequence?
Look back to thedefinition of
bounded
and
unbounded
.
Rather unexpectedly, it is possible to show that
any
sequence has a
monotonic subsequence. To prove this we will use the notion of a
floor
term
of a sequence. We will call a term of a sequence a 'floor term'
when none of its successors is strictly less. The term
a
is a 'floor term'
if
a
when
n
f
. A 'floor term' is a lower bound for the rest of the
sequence. The phrase 'floor term' is used in questions 15 and 16, and
not again in thecours.
a
