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(iii) As (i) with upper bound max(
a
,
a
,...,
a
,
U
) and lower bound
min(
a
,
a
,...,
a
,
L
).
14 (a) No. Take the sequence (
a
) defined by
a
1 and
a
n
, for
example. (b) (i) Yes,
a
0,
a
(
2)
; (ii) no, (
2)
.
15 (i)
1,
1/3,
1/5, . . .; (ii) 0, 0; (iii) none.
succeeds the floor term
a
, then by the
16
(i) If thefloor trm
a
definition of
a
,
a
a
, and the subsequence (
a
)is
monotonic increasing.
(ii) Since no term after
a
is a floor term, every term thereafter is
eventually succeeded by a lesser term, giving a strictly
decreasing subsequence.
(iii) Since there are no floor terms, every term is eventually
succeeded by a lesser term, again giving strictly decreasing
subsequence.
17
(i) Not monotonic; bounded below; all terms
1.
(ii) Not monotonic; unbounded.
(iii) Not monotonic; bounded below; all terms
1; all terms past
the10th
10; all terms past the 100th
100.
(iv) Not monotonic; bounded; all terms
10.
18 (i) 1
C
; (ii) any
C
; (iv) 11
C
.
19 (i)
A
1
[
A
]
A
. (ii) (a) 0
x
N
(
x
)
1 which is impossible
when
x
is an integer. (b)
x
M
(
x
1)
x
1, so not possible
when
x
is an integer. (c) Of course
K
(
x
)
[
x
] is possible, and is
inevitable when
x
is not an integer, but when
x
is an integer we
may have
x
K
(
x
)or
K
(
x
1).
20 Take
N
C
.
21 (a) Take
N
C
/
y
.
22 Although in the common-sense use of the word, a sequence which
tends to
is 'increasing', qn 17 (iii) shows that such a sequence
need not be monotonic increasing, and therefore not, in the
mathematically precise sense, increasing.
23 (
a
)
when, given any number
C
, there exists an
N
such that
n
N
a
C
.
24
(a) Any negative number is a lower bound; also 0.
(b) Any positivenumbr
1 is an upper bound; any positive
number is eventually an upper bound.
25 First sequence: answers as in qn 24.
Second sequence: any positive number is an upper bound; any
