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are unbounded tend to
. Questions 34 and 35 prove that the
convergence of infinite decimal sequences implies the convergence of
monotonic bounded sequences. The converse is obvious, since every
infinite decimal sequence is bounded and monotonic. So we could have
adopted the convergence of monotonic bounded sequences as our
completeness principle.
36 Use qn 2.49 to prove that the sequence with
n
th term (1
1/
n
)
has
a limit which lies between 2 and 3.
nth roots of positive real numbers, n a positive integer
37 If
a
and
x
are positive real numbers with
a
x
, provethat
a
(
(
x
a
/
x
))
x
. Deduce that if
x
is positiveand
a
x
, the
sequence (
x
) defined by
is convergent. Find its limit. Deduce that every positive real
number has a unique positive real number as square root. Compare
with qn 2.37. (This method of approximating to square roots is
usually known as Horner's method, though it coincides with
Newton's approximation in appendix 3, and was used centuries
before by the Chinese.)
1
2
a
x
x
x
38 Prove that the sequences (
a
) of question 2.38 are both
convergent to the same limit. Prove the analogous result for
question 2.39.
) and (
b
39 Use question 37 to prove that positive real numbers have real 4th
roots, 8th roots, 16th roots, etc.
40 Let
k
be a positive integer.
If 0
c
1, then
c
c
1
, and if 1
c
, then 1
c
c
,byqn
2.11.
k
0
c
1
x
k
x
0
c
1
c
k
1
x
k
x
1
c
