Graphics Reference
In-Depth Information
A sequence which is bounded above and
bounded below is said to be
bounded
.
Definition
qn 8
A sequence (
a
) is called a
subsequence
of (
a
), if
n
n
whenever
i
j
.
T
heorem
qn 16
Every sequence has a monotonic subsequence.
Definition
qn 18
A sequence (
a
) is said to
tend to infinity
if and
only if for any number
C
, there is an
N
, such
that
n
N
a
C
. This is written (
a
)
as
n
.
Property of Archimedean order
Given a number, there is an integer which is
greater. Equivalently, the positive integers are
not bounded above.
Theorem
For every real number
A
, there is a unique
integer [
A
] such that [
A
]
A
[
A
]
1.
Null sequences
Even to the untutored eye, the sequence
1,
,
,
, ...,
,...
will be seen to 'tend to 0'. We work towards a precise definition of this
phrase and its synonym 'converges to 0'.
24
(a) Name some lower bounds for the sequence (
).
Are all possible lower bounds negative?
(b) Name some upper bounds for the sequence (
).
Are all possible upper bounds positive?
Are all positive numbers upper bounds for the sequence?
Is each positive number
eventually
an upper bound for the
sequence?
It seems natural to say that a sequence like
,
,
,
,...,
,...
tends to 1, or that a sequence like
; the
first from below, the second from above. In both cases, the limit is not
reached. The 'one-sidedness' of the terms and the non-reaching of the
limit are part of a common-sense notion of 'tends to'. The examples of
convergence explored during the eighteenth century normally had both
these properties. But in order to reach a good definition of convergence
today, our attention will be drawn to bounds and to nearness, and with
this shift of attention neither the reaching of a limit, nor an oscillation
about a limit, is of concern.
,
,
,
, ...,
, . . . tends to
