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This proves that every infinite bounded set has a cluster point. It was
formulated by Weierstrass about 1867, and proved by a method due to
Bolzano (1817) of repeated bisections of the bounded interval
containing thest
A
, choosing always a half containing an infinity of
points of
A
. Thenam, Bolzano
—
Weierstrass, was given by Cantor
(1870) and Heine (1872). Cantor said that this theorem was the basis of
all themoreimportant mathematical truths.
To understand the significance of this theorem it is necessary to
recognise that it would be false if the rational numbers were the only
ones available.
54 Let
A
be the set of terminating decimals in the infinite decimal
sequence for
2. State integer bounds for this set. Prove that no
rational number can be a cluster point for
A
.
Cauchy sequences
The great virtue of the theorem that a bounded monotonic
sequence is convergent is that it provides a method of determining the
fact of convergence without knowing the value of the limit.
Is there a condition guaranteeing the convergence of a
non-monotonic sequence which does not require knowledge of the
valueof thelimit?
A reasonable conjecture would be to suggest that a sequence was
convergent if the differences between consecutive terms form a null
sequence. But there are counter-examples to such a conjecture. One is
provided by the harmonic series which we will examine in the next
chapter. Another derives from the sequence (
n
).
We have shown that the differences between consecutive terms
(
(
n
1)
n
) gives a null sequence in qn 3.35(v), and that the
sequence itself tends to infinity in qn 3.20. So the condition '(
a
a
)
is null' does not guarantee the convergence of (
a
).
a
) for someconstant
k
is
no more effective, even for a large value of
k
, than for
k
1 sincethe
sequence (
Examining differences of the form (
a
(
n
/
k
)) also tends to infinity. However, it is worth examining
the hypothesis that the difference (
a
) is sandwiched between two
null sequences for
all
values of
k
. If for example
a
1/
n
a
1/
n
, for all positive integers
k
,
is this suMcient for the sequence (
a
a
) to be convergent? We first
establish the converse result.
