Graphics Reference
In-Depth Information
Bounded monotonic sequences
33 Explain why every monotonic sequence is either bounded above or
bounded below.
Deduce that an increasing sequence which is bounded above is
bounded, and that a decreasing sequence which is bounded below is
bounded.
34 (after
du Bois
-
Reymond
, 1882) Wesupposethat (
a
) is a monotonic
decreasing sequence which is bounded below, by
L
, and we seek to
provethat (
a
) is convergent.
) which is an integer.
(ii) Name an upper bound for the sequence (
a
(i) Name a lower bound for the sequence (
a
) which is an
integer.
(iii) Must there be consecutive integers
c
,
c
1 for which
c
is a
lower bound for (
a
) and
c
1 is not? For such integers let
c
.
(iv) Consider the eleven numbers
t
t
d
,
t
,
t
,...,
t
.
c
10
Is there a
c
0, 1, 2, . . ., or 9 such that
t
c
1
is a lower bound for (
a
) and
t
is not?
10
.
(v) Proceed inductively to define an infinite decimal sequence (
t
Let
d
c
and
t
d
.
d
),
in such a way that
t
is a lower bound for the sequence (
a
)
while
t
1/10
is not a lower bound.
Define
t
c
10
t
, with
c
0, 1, 2, . . ., or 9.
where
t
is a lower bound for (
a
) while
t
1/10
is
not.
c
d
, and
t
d
.
d
d
d
...
d
d
.
) convergent? Let its limit be
D
.
(vii) For a constant
k
, say why (
a
(vi) Why is the sequence (
t
t
a
D
)
0.
(viii) How do you know that for suMciently large
n
,
D
a
D
1/10
for a given positive integer
i
.
(ix) Provethat (
a
)
D
.
35 Use question 34 to prove that a monotonic increasing sequence
which is bounded must be convergent.
Now we have established that monotonic sequences which are bounded
are convergent, and it easily follows that monotonic sequences which
