Graphics Reference
In-Depth Information
n
15 Underline the floor terms in each of the following sequences.
(i)
1,
,
,
,
,
,...,(
1)
/
n
,...
,...
(iii)
1, 2,
3, 4,
5, 6, . . ., (
1)
n
,...
(ii) 1, 0, 1, 0, 1,
,1,
,1,
,...,1,
Sketch a graph of the first six terms of each of these sequences and
circle the points corresponding to floor terms.
16
(i) If an arbitrary sequence contains an infinite number of floor
terms, show that they form a monotonic increasing
subsequence.
(ii) If an arbitrary sequence contains just a finite number of floor
terms, and the last one is
a
, form a strictly decreasing
as its first term.
(iii) If an arbitrary sequence contains no floor terms at all, form a
strictly decreasing subsequence with
a
subsequence with
a
as its first term.
The theorem proved in qn 16 guarantees the existence of a monotonic
subsequence even for a sequence such as (sin
n
).
a
n
1
sin
n
0.5
n
0.5
1
