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Thus for any positivenumbr
c
1, wecan choosepositive
numbers
a
and
b
such that
a
c
b
.
Now consider the number
d
)/2.
If
d
c
, wehavefound a
k
th root of
c
.
If
d
(
a
b
c
,let
a
d
and
b
b
.
If
d
c
,let
a
a
and
b
d
.
So
a
a
b
b
,
a
c
b
, and
b
a
(
b
a
)/2.
Repeat this process to define sequences (
a
), (
b
) inductively such
that
a
a
b
b
,
a
c
b
, and
b
a
(
b
a
)/2.
(i) Why must the sequence (
a
) be convergent?
(ii) Why must the sequence (
b
) be convergent?
(iii) Why must (
b
a
) be a null sequence, and the limits of (
a
)
) equal?
(iv) Denote the common limit by
l
. Why must (
a
and (
b
)
l
and
l
?
(v) Why must (
b
(
b
)
a
) be a null sequence?
c
b
and the squeeze rule to prove that (
c
a
(vi) Use
a
)
is a null sequence.
(vii) Deduce that
c
l
, so that
c
has a
k
th root, namely
l
.
(i) Prove that the sequence (
n
(
a
1)) of qn 2.50 is convergent
when 1
41
a
.
(ii) Provethat lim
n
(
(1/
a
)
1)
lim
n
(
a
1), using qn 3.57
and thequotient rule3.67.
(iii) Provealso that lim
n
(
(
ab
)
1)
lim
n
(
lim
n
(
a
1)
b
1), using qn 3.57 and theproduct
ruleqn 3.54(vi).
If you consider lim
n
(
a
1) as a function of
a
, can you think of
any other function of
a
with similar properties?
Nested closed intervals
42
The Chinese Box Theorem
(
a
) and (
b
) are sequences such that
a
a
b
b
for all
values of
n
.
Let [
a
,
b
] denote the closed interval
x
a
x
b
.
].
We describe this by saying that these intervals are
nested
.
(ii) Why must each of the two sequences be convergent?
(iii) By considering the limits of the two sequences prove that at
least one number lies in all the intervals [
a
(i) Say why [
a
,
b
]
[
a
,
b
]
...
[
a
,
b
]
[
a
,
b
,
b
].
