Graphics Reference
In-Depth Information
A sequence (
a
) is said to bea
null sequence
, or to tend to 0, or
converge to 0, when,
for any positivenumbr
, however small, there is a stage in the
sequence beyond which the remaining terms of the sequence are
between
and
;
or, equivalently, for any given positive
,
a
n
ε
n
0
ε
is eventually an upper bound for the sequence and
is
eventually a lower bound for the sequence:
or, equivalently,
all but a finite number of terms of the sequence lie between
,or
satisfy
a
.
These conditions must hold for every possible choice of
. So, to be
called a null sequence, the sequence must satisfy an infinity of
conditions.
Formally we write (
a
n
)
0as
n
if and only if, given
0,
there is an
N
such that
n
N
a
n
.
This definition could also have been given by saying that the
subsequence of all terms after
a
is bounded by
. If a sequence is
null, there are many acceptable
N
s for each
.
28 For each of the sequences of qn 26, prove that it is not null by
naming an
for which no related
N
exists. What about the
constant sequence defined by
a
1? See Fig. 3.28
29 Prove that the sequence (1/
n
) is a null sequence. Use the
Archimedean order property for the proof.
