Graphics Reference
In-Depth Information
or generally, the strictly increasing sequence
n
,
n
,
n
,
n
, ...,
n
, ...,
all provide examples (the last is a thoroughly general example) of
infinite subsets of the natural numbers, N.
With any such infinite subset, a subsequence of a sequence may be
constructed.
Thus from the sequence
a
,
a
,
a
,...,
a
, ...
various subsequences, such as
a
,
a
,
a
,...,
a
,...
a
,
a
,
a
,...,
a
,...
a
,
a
,
a
,...,
a
,...,
may beconstructed (using thefirst, thethird and thesixth of the
subsets of N given above), or in general
a
,
a
,
a
,...,
a
,....
You should think of
n
as thefirst suMx (or valueof
n
) used in the
subsequence,
n
as the second suMx used in the subsequence, and so on.
One erases some (or none!) of the terms of a sequence to obtain a
subsequence.
A
subsequence
of a sequence (
a
with the
terms of the subsequence occurring in the same order as in the original
sequence.
In the definition of a sequence (
a
) is an infinitesubst of the
a
),
n
runs through all of thenatural
numbers,
, in order.
With the general notation for a
subsequence
(
a
N
runs through some
(or all) of the natural numbers in order (and
i
runs through them all).
),
n
8 Why are none of the following sets, as they stand, subsequences of
(
a
)?
(i)
a
,
a
,
a
,
a
,
a
,
a
,
a
,
a
,
a
,
a
;
(ii)
a
,
a
,
a
,
a
,...,
a
,
a
,...;
(iii) (
a
);
(iv) (
a
).
9 For the sequence (
a
)
10,
11,
12, 13, 14,
15,
16, 17, 18,
19,
20, 21, 22, . . . identify
n
,
n
,
n
,
n
and
n
for
(i) the subsequence of positive terms;
(ii) the subsequence of negative terms;
(iii) the subsequence of even terms;
(iv) the subsequence of odd terms.
