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6.19
Definition ( converges to .)
Let be a sequence of real numbers, and let be a real number. We
say
that converges to if for every positive number
there
is a number in
such that all of the numbers for
which
approximate with an error smaller than . We
denote the fact that converges to by the notation
Thus ``
" means:
For every
there is a number in
such that
Since
it follows immediately from the definition of convergence that
We will make frequent use of these equivalences.
6.20
Example.
If
then
Proof: Let be a generic element of
. I must find a number
such that

(6.21) 
whenever
. Well, for all
in
Now for every
in
we have
and by the Archimedean property of
there is some integer
such
that
. For all
we
have
so by (
6.22)
Hence by the definition of convergence we have

(6.23) 
A very similar argument can be used to show that

(6.24) 
6.25
Example.
In the eighteenth century the rather complicated argument just
given would have been stated as
If is infinitely large, then
.
The first calculus text book (written by Guillaume François
de l'Hôpital
and published in 1696) sets forth the postulate
Grant that two quantities, whose difference is an infinitely
small quantity, may be taken (or used) indifferently for each other:
or (which is the same thing) that a quantity which is increased or decreased
only by an infinitely small quantity, may be considered as remaining
the same[35, page 314].
If
is infinite, then
is infinitely small, so
, and similarly
.
Hence
There were numerous objections to this sort of reasoning. Even though
, we do not have
, since
It took many mathematicians working over hundreds
of years to come up with our definition of convergence.
6.26
Theorem (Uniqueness theorem for convergence.)
Let be a sequence of real numbers, and let be real
numbers. Suppose
Then .
Proof: Suppose and . By the triangle
inequality

(6.27) 
Let be a generic element of
. Then
is also in
. Since , there is a number
in
such that

(6.28) 
Since there is a number
in
such that

(6.29) 
Let be the larger of
and
. If is a positive integer
and
then by (6.27), (6.28), and
(6.29), we have
Since this holds for all in
, we have .
6.30
Definition (Limit of a sequence.)
Let
be a sequence of real numbers. If there is a number
such that
, we write
. The uniqueness
theorem for convergence shows that this definition makes sense. If
, we say
is the limit of the sequence .
6.31
Definition (Convergent and divergent sequence.)
Let
be a sequence of real numbers. If there is a number
such that
, we say that
is a
convergent
sequence. If there is no such number
, we say that
is a
divergent sequence.
6.32
Example.
It follows from example
6.20 that
for all
in
. Hence
is a convergent sequence
for each
in
.
The sequence is a divergent sequence. To see this, suppose there
were a number such that .
Then we can find a number
such that
In particular
(since
is an integer
greater than
).
Hence, by the triangle inequality
i.e.,
which is false.
Since the assumption has led to a contradiction, it is false
that .
6.33
Exercise.
A
Let
be a sequence of real numbers,
and let
be a real number. Suppose that as
gets larger and larger,
gets nearer and nearer to
, i.e., suppose that for all
and
in
Does it follow that
converges to
?
Next: 6.4 Properties of Limits.
Up: 6. Limits of Sequences
Previous: 6.2 Approximation
Index
Ray Mayer
20070907