Real Analysis 1 & 2 Notes

Analysis I
Definition The of f is the
What
.
imap
isAnalysis ? Set

(b B) f(a)
- -




Analysis is the careful study of (m(f) = + = b for a + A3
functions and the items and spaces
we
pluy in and
get out of them.

Definition. A set is ordered if Va b , CES ,


For items in the space questions ,
·
alb a = b ,
or asb
,

b
revolve around position , distance magnitude, a < bx = ax
·

, ,




proximity , orthogonality ,
bases ,
i More.

Theorem. D is the smallest ordered
For functions , questione look like : field which contains N.
Continuity
·




·

Differentiability Remark .
VXeQ ,
UneN ,
J yeD such
·


Integrability that(x-ykh.

Example of
something neither real nor Remark. For x
, yeQ (31) (x y/ -




complex : is the distance between them .




Let 23 (0 13) ,
= Ef : Co 13 ,
>
-
R) Definition (kind of)
. .
R is the smallest
ordered field Q with
containing
:
no


( % dx =
3 "holes. "


Define (3(50 13)
E :
,
>
- I such Definition D .
=
IRIQ is the set of
that F(f) is the that
energy required
real numbers are not rational ,
i . .
e


to sustain a
physical phenomenon in irrational.
an .
environment

Remark. It is the set of holes in

When F is minimized we know the D .


,


behavior of the system .




Remark. The actual definitions
construction of IR is in
chapter
Definition .
Let A ,
B be sets. 8 of Abbott .




A function F: A - B is a relation
that VatA a
unique
assigns
be B fla) ,
= b
.


Definition .
For F A is the
,

domain and B is ,
the codomain .

,Theorem Let a belR
.




,| a b |<| a| +(b)
+
,
In general, a b = -
b <ab
2 Ilal-1b/l|a bl - 2
asb1 asb = lakb
3 labl lallbl=




Theorem .
Let a
,
belR .
Then
,
a = b if

Proof . and only if VE-O ,
la-bid
?
I
Consider la obl (a ob)=




=
a 2ab +b If a = b la- bl = s Vaso .
, ,
= lak o Cabo lb1
Notice ab < lab) lallbl = slaks2allblolb1 Example .
3 , 3 16 .

,
3 * 3 .
1 .




=
((a)p(b))2
5 =
2 ,
13-3 112 .




Thus la obl< (1a)o(b1)" and
, ,
2 =
1
,
13-3 11 .




laob/slab/b) because both ,
side s =
E ,
13 -
3 .
11
as a tire Il 2 = 0 1 13- 3 11 =
0 1
non-my
. .
.
, .



-


I hus ,
3 * 3 1..




2 Notice (x15y()-y > Xy
b)
,

=
.




Since labbIsIa/b/b1 choose ,
a= X
,
Suppose a b
,
let 3
bl
and b =
y - X, then la-bk < ,
this is
impossible .


Thus a = b 1/5)

(y x))
.
,



(x + -
> (x | +|y x) -




-
(x) -

|x| Remark .
(a blc(=)
-
-
Ea -
b

(y| (x | =
-



/y x| -
-
2(a -
b2d

-
Ebb (a<Ebb

(yp(x y)) -(y)p(y x)
- -

b -
EacbbE


(a) (b) =(b al Essentially we're to
looking
- -




,

Without loss of generality through a window of radius a

around a, andf b is inside
(a) (b) = 0
- = /(b) Call -
= (a)-1b) the window no matter how small
Ilal-Ibll
,

= &
gets , we conclude that they
=
(a)-Ib/ must be the same number .




As la(Iblslb- al
=> Ila) 1 bl)s | a b /
-
-
11]

,Completeness
Definition . Let SSIR .
XeS is the maximum

Definition . A set SCR is bounded of S if for all yeS yeX Denoted ·

,


above of JMEIR such that , for all max(S) = X .
Likewise we define the
XIR XM . minimum of S , denoted min(s) = zeS if
,




XyeS , 754 .




S is bounded below ifINEIR
such that for all xeIR X = N .
Definition .
A set SER has the least
,


upper bound property if for all ASS,
S is bounded if JPCR such that J sup(A) eS .




for all xeIR (x1sP
,




Example
. A = 10 13 u (2 3) v54-IneN3
, ,
Axiom of Completeness. R is an ordered
field
satisfying the suprmum property.
A by 4 5 6
is bounded above , ... , ,

A is bounded below by 0
,, 1 ...
Lemma Let xelR be
. an
upper
bound
for ASIR Then . x= sup(A) if and

only if VE3O JacA such that
,


Definition .
Let SER if S is bounded X -

Ea
,


above by M
,
then M is an
upper
bound of S
.
pf .
=
) Suppose not . Then JEc0
such that VacA X-Ea . Then
,

The least upper bound of S (if it a -
E<X is an upper
bound of A
,
i . .
e



exists) is
XR such that a is not the the supremum.
,


upper bound of S
X is an

2 Vo that are
upper bounds of S
, # Let 1 = supA .
Then since X is

XSr an
upper bound for A , 154 .

We
say
thatX is the supremum of Further ,
as VECO ,
Jaf A s .
t .




S ,
denoted sup(s) = X .
X- Ea
,
x- ECVEs8 .
Thre ,



OX-1CE and X-1 = 0 .
TJ
,


Likewise there exists a greatest
,



lower bound defined
correspondingly, Remark. All significant properties of supremums
we say
that the greatest lower bound have natural
correspondences with
of S B is the infimum of S
, ,
, propertice of infirms.
denoted inf(s) B
. =

, Theorem . Let o be a lower bound of ASR ,
Question : Is N bounded above in IR?
then
p-inf(A) if and only if
V S>0 Jae A such that Bossa .
Example .
Consider the Geld Q(X) .

,


RsD(x) ,
Moreour R(X) is ordered


An open interval in IR
F
Definition. the
is a if
leadingofficea
subset of IR of the form
of fi(x) g (x) -
G (x)g , (x) is positive
x
. ,



positive .




C =
Ex eR(bxxca] =
(b a) ,
We see R is thus bounded
above by X but there is no
,

least upper bound.
And a closed interval ,
likewise



x Theorem
IR
.


.
I is not bounded above in




c =
Ex eR(bxxca3 =
(b a] ,



pf Assume FaeR
.
such that x > nV nel
.
Theorem .
Letbut an such that an and ,
Further let x =
sup[N). Then there
bnfbu bn ,
=
am Un me N
,
exists nel such that X-1n .


Then x < nol x * Sup(N) == $


In [an bu]
, .




Let =

,
then In Fro
VneX Let XeR then Jnel
.

Grollary .


,
such

that x<n .

Further In D
Corrollay Let yeR > 0
. Then Ine *
SaulneM]
.

, y
Proof .
Let A =
,
clearly such that %
A is bounded above by br Unex
.
,



Let x sup(A) Then asbu Uncl
=
.


, ,
Definition .
A set AsR is dense in

but then f In VneN and IR if XX yelR x<y , such that
, , ,



a TJ JaeA where xcacy ·




What about In-Lan bn) ? ,




(0 h) 0 . 0 ,
=
,
10) ?

10, De 10 10h) . ,
,