Graphics Reference
In-Depth Information
58 Locatethecontinued fractions
1
2
1
1
1, 1
,1
,1
,1
,
1
2
1
2
2
1
2
2
on a number line.
If, for all
n
, the terms of a sequence (
a
) satisfy the inequalities
a
a
a
a
, and
a
a
1/
n
, provethat the
sequence is convergent.
Least upper bounds (sup) and greatest lowest bounds (inf)
Upper bounds and greatest terms
59 Some sets of numbers have a greatest member.
What is the greatest member of
0, 1, 2, 4,
100,
50
?
Must any finite set of numbers have a greatest member?
What is the greatest member of
x
0
x
1
?
x
x
60
(i) (a) Is 1 an upper bound for
?
(b) Can any number less than 1 be an upper bound for
x
0
1
?
(ii) (a) Is 3 an upper bound for
x
0
x
1, 2
x
3
?
(b) Can any number less than 3 be an upper bound for
x
0
x
1, 2
x
3
?
0
x
1
61 Check that 2 is an upper bound for each of the following sets:
(i)
x
0
x
1
,
(ii)
x
x
0
1
,
(iii)
x
0
x
,
x
1, 1
x
1
,
(iv)
x
0
x
,1
x
1
,
(v)
1
1/
n
n
N
,
(vi)
2
1/
n
n
N
,
(vii)
1
(
1)
/
n
n
N
,
(viii)
x
x
2,
x
Q
,
(ix)
q
q
2,
q
Q
.
For which of these sets can you find a number less than 2 which is
still an upper bound for the set?
