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, provethat
a
b
. Makea proof by
contradiction. Usetrichotomy and qustion 14.
a
,0
b
and
a
b
17 If 0
A consequence of this is that if 0
a
b
, then 0
a
b
. Notethat
(
a
) is always positive, or zero.
18 Sketch a graph of
y
x
and illustrateon it how
a
b
may be
compatiblewith any oneof
a
b
,
a
b
and
b
a
.
19
(i) With an argument like that of question 14, show that if
0
a
b
, then
a
b
.
(ii) With an argument like that of question 17, show that if 0
a
,
0
b
and
a
b
, then
a
b
.
a
b
, proveby induction that
a
b
20
(i) If 0
for all positive
integers,
n
.
(ii) If 0
for some positive integer
n
,
provethat
a
b
. A consequence of this is that if 0
a
b
,
then 0
a
b
.
a
and 0
b
, and
a
b
21 Sketch graphs of
y
x
for various positive integers
n
, and decide
for which positive integers
na
b
a
b
.
22 If 0
a
, provethat 0
1/
a
. Usequstions 12 and 16.
23 If 0
a
b
, provethat 0
1/
b
1/
a
.
24 If
a
0, provethat 1/
a
0.
25 If
a
b
0, provethat 1/
b
1/
a
0.
26 Sketch a graph of
y
1/
x
and illustrate on it how, provided neither
a
nor
b
is 0,
a
b
may be compatible either with 1/
b
1/
a
or with
1/
a
1/
b
.
27 Sketch a graph of
y
1/(1
x
). Decide for what values of
x
,
1/(1
x
)
1, and then prove your claim formally.
a
b
is defined to mean
either a
bora
b
. Thus both 2
3 and
2
2 are true. Likewise
a
b
is defined to mean
either a
bora
b
.
28 Trueor fals?
(i)
a
b
a
b
1.
(ii)
a
b
a
b
1.
(iii)
a
b
a
b
.
(iv)
a
b
a
b
.
29
Bernoulli
'
s inequality
Usequstion 11 to proveby induction that if
1
x
, then
1
nx
(1
x
)
for all positive integers
n
.
For positivevalus of
x
, this can also be deduced from the
