Real Analysis
,eek 1
#
der
Relation als
Recap e
-
commutativity [a + b = b+ a ,
ab = ba] -
·
·
Associativity [(a + b) +c = a + (b + ) ,
a(bx) =
(ab)9 a
,
b :
. b b
ac] a
Distributivity
ab
·
b
a
=
=
)
=
[a(b
a
·
+
=
+
a 1) If u
= b and b= then
Inverse [a 0 C
-
+
=
· -
a =
n
,
(transitive)
Identity
a c
[a + a)
=
·
o = a 1 ·
a =
,
#f a= b then at c = b + c
Field : a set with two
operations (X ,+ and for and
any c
=C
then
all reals
above
properties e
.
g
.
,
rationals
accb
(a) =
Ja a = o ordered field >
-
a field
1 -
aa = 0
with an order relation
, Frequala b
Triple
Supremum B SupS S /R
·
>
-
= :
>
7) b E b
: ec =
,
VaES"
B -
>
no numberless than B is
↳ la-bl = llal-Ibll and la + bl = llal-Ib/l
an
upper bound of S
b -
an
upper bound but there
A sel bounded if there numbers and
greater upper bounds
is are a can be
b St a ExEb EaES
Infimum infs
.
·
> a
-
=
↳
infsESupS 10 1 % -
finite
Jat
,
y = FyCS
: a
to 13 infinite
,
not bonded
>
11 0)
-
,
>
-
, - >
-
no number more than X is
above
bounded below a lower bound of S
not bounded a >
- a lower bound could be
,
smaller lower bounds .
(not
uniques
, rationals and irrationals
completenessxioma
The above
tensity of
~
&: A set D is dense in the reals
then it has a supremum if every open
interval (a
,
b) contains a
The real system is
member of
a
D
complete ordered field
however
not
,
the rational field is
notiationals
The are dense in the
reals and b real numbers
; if a are
I a nonempty set s of real
with
↓ S
a=
t
b
act = b
there is an
irrational number
numbers bounded above the
. .
is
proof rationals dense in IR there
the unique Since
: are
real number B st
,
supS
is .
are rationals r , 12 S . t .
a = = re = b
CEB EetS
t
+ E(k - r)
·
Let =
r >
-
- Q
·
if 3 = 0
,
7920ES : <Co = B-E Then t is irrational and n = t =
ve
,eek 1
#
der
Relation als
Recap e
-
commutativity [a + b = b+ a ,
ab = ba] -
·
·
Associativity [(a + b) +c = a + (b + ) ,
a(bx) =
(ab)9 a
,
b :
. b b
ac] a
Distributivity
ab
·
b
a
=
=
)
=
[a(b
a
·
+
=
+
a 1) If u
= b and b= then
Inverse [a 0 C
-
+
=
· -
a =
n
,
(transitive)
Identity
a c
[a + a)
=
·
o = a 1 ·
a =
,
#f a= b then at c = b + c
Field : a set with two
operations (X ,+ and for and
any c
=C
then
all reals
above
properties e
.
g
.
,
rationals
accb
(a) =
Ja a = o ordered field >
-
a field
1 -
aa = 0
with an order relation
, Frequala b
Triple
Supremum B SupS S /R
·
>
-
= :
>
7) b E b
: ec =
,
VaES"
B -
>
no numberless than B is
↳ la-bl = llal-Ibll and la + bl = llal-Ib/l
an
upper bound of S
b -
an
upper bound but there
A sel bounded if there numbers and
greater upper bounds
is are a can be
b St a ExEb EaES
Infimum infs
.
·
> a
-
=
↳
infsESupS 10 1 % -
finite
Jat
,
y = FyCS
: a
to 13 infinite
,
not bonded
>
11 0)
-
,
>
-
, - >
-
no number more than X is
above
bounded below a lower bound of S
not bounded a >
- a lower bound could be
,
smaller lower bounds .
(not
uniques
, rationals and irrationals
completenessxioma
The above
tensity of
~
&: A set D is dense in the reals
then it has a supremum if every open
interval (a
,
b) contains a
The real system is
member of
a
D
complete ordered field
however
not
,
the rational field is
notiationals
The are dense in the
reals and b real numbers
; if a are
I a nonempty set s of real
with
↓ S
a=
t
b
act = b
there is an
irrational number
numbers bounded above the
. .
is
proof rationals dense in IR there
the unique Since
: are
real number B st
,
supS
is .
are rationals r , 12 S . t .
a = = re = b
CEB EetS
t
+ E(k - r)
·
Let =
r >
-
- Q
·
if 3 = 0
,
7920ES : <Co = B-E Then t is irrational and n = t =
ve
