Graphics Reference
In-Depth Information
negative
a
/
b
b
a
1
2
3
4
1
30
60
120
240
2
90
180
360
720
3
270
540
1080
2160
4
810
1620
3240
6480
Sequence of non-zero rationals:
,
,
,
,
, [2/2],
,
,
,
,
,
,
...
26 1
2, 2
2,
2, 3
2, 1
2
2, 1
2, ...
27
Z
,
T
,
D
, and
Q
are countably infinite. The set of infinite decimals is
not.
28 All of
N
is deleted. The total length removed is 2 units.
29 All of
N
is deleted. The total length removed is 2
units.
may beas
small as wewish. So
N
occupies no length.
30 All of (
a
) is deleted with a total length of 2
units. Thelength is
arbitrarily small, so a sequence (a countably infinite set of numbers)
does not occupy length on the line.
Sincetherationals arecountably infinit, so aretheinfinitedcimals
equal to rationals between 0 and 1. If a sequence of these infinite
decimals was used the new decimal constructed would be irrational.
31 The sequence is monotonic increasing. Every term is a terminating
decimal and therefore rational. 0.2, 0.13, 0.124, 0.1235 etc. are all upper
bounds for the sequence.
32 Apply thesum rule(3.54(iii)), thediffrencerule(3.54(v)), theproduct
rule(3.54(vi)) and thequotient rule(3.67) to theinfinitedcimal
sequences.
33 The first term of an increasing sequence is a lower bound. The first
term of a decreasing sequence is an upper bound.
34
1.
(iii) Yes. Observe the finite number of integers between (i) and (ii).
(iv) Since
t
(i) [
L
].
(ii) [
a
]
is a lower bound and
t
1 is not, wecan argueas in
(iii).
(v) Argueas in (iii).
(vi) (
t
) is an infinite decimal sequence which converges by the
completeness principle.
(vii) Thelimit is
a
D
by the difference rule (3.54(v)). Since
t
is a
