Graphics Reference
In-Depth Information
13 Can you givemeaning to thefollowing:
1
(i)
1
,
1
2
1
3
4
...
(ii)
(1
(2
(3
. . .))),
(iii) 2
2, 4
(2
2), 8
(2
(2
2)),
16
(2
(2
(2
2))), . . .
k
,
14 (
Cauchy
, 1821) If for a sequence of positive terms (
a
/
a
)
a
provethat (
)
k
. What about the converse?
15 (
Cantor
, 1874) A real number is said to be
algebraic
if it is the
solution of a polynomial equation with integer coeMcients:
a
a
x
a
x
...
a
x
0.
Calling
a
n
, the'wight' of the
polynomial show that the set of algebraic numbers is countable. A
real number which is not algebraic is said to be
transcendental
.
a
a
...
a
16 Does a real sequence (
a
) necessarily converge if, given
0, there
exists an integer
N
(depending on
p
and
), such that
a
a
when
n
N
, for each integer
p
?
17 Is there a non-empty set of real numbers which is both closed
(contains all its limit points) and open (contains a neighbourhood
of each of its points)?
18 Under what circumstances does
lim
(
ax
bx
c
)
(
Ax
Bx
C
))
(
exist?
19 (
Pringsheim
, 1899) Find thevalus of thefunction
f
defined by
f
(
x
)
lim
lim
(cos
n
!
x
)
.
20 If
a
,
b
,
c
and
d
areirrational numbrs with
a
b
and
c
d
, can
you construct a bijection with domain [
a
,
b
]
Q and range
[
c
,
d
]
Q
? A continuous bijection? A differentiable bijection?
