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a
, usethefact that
55 If (
a
)
a
a
a
a
a
a
,
from thetriangleinequality, qn 2.61, to show that
may
be arbitrarily small, irrespective of the value of
k
, provided only
that
n
is largeenough.
Or, to use more precise language, that given
0, there exists an
N
such that
a
a
a
a
, when n
N
.
So the condition we are examining is a necessary consequence of
convergence. The argument in qn 55 does not depend upon
completeness.
The condition we are investigating is called the Cauchy criterion.
A sequence (
a
) is said to satisfy the Cauchy criterion, when
given
0, there exists an
N
such that
n
N
a
n
a
n
, for all positive integers
k
.
k
A sequence satisfying the Cauchy criterion is called a Cauchy sequence.
In question 55, we showed that every convergent sequence is a Cauchy
sequence. We now explore the converse.
a
a
56 If (
a
) is an infinite decimal sequence prove that
1/10
.
This shows that an infinite decimal sequence is necessarily a
Cauchy sequence.
Thus the claim that every Cauchy sequence is convergent implies
the completeness principle.
57
The General Principle of Convergence
.
Let (
a
) be a Cauchy sequence of real numbers.
By putting
1 in the Cauchy criterion prove that every Cauchy
sequence is bounded (use qn 3.13). Use question 46 to establish the
existence of a convergent subsequence (
a
a
.
)
Now usethetriangleinequality
a
a
a
a
a
a
to
provethat (
a
a
.
So every Cauchy sequence of real numbers converges to a real
limit.
)
In 1882,
P
.
du Bois
-
Reymond
gave the name 'The General Principle of
Convergence' to the proposition that a sequence is convergent if and
only if it is a Cauchy sequence.
