Real Analysis (Chapter-1)

Analysis:
Definition:Mathematical analysis is the branch of mathematics which deals with limits and related theories such as differentiation,integration,infinite series and analytic Function. It is of two types Ex:(i)Real Analysis (ii)Complex Analysis
Real Analysis:
It is the study of sequence and their limits,continuity,differentiation,integration and sequence of function.It deals with the function of real variables.
Field:
A field is a set ' F ' with two binary operation denoted by ' + ' and ' . ' and must satisfy the following axioms.

[Properties] [Condition] Ex:[Addition] [Multiplication]

1.Closure	∀a,b∈F	a+b ∈F	a.b ∈F
2.Commutative	∀a,b∈F	a+b=b+a	a.b=b.a

3.Associative	∀a,b,c∈F	a+(b+c)=(a+b)+c	(a.b).c=a.(b.c)
4.Identity	∀a∈F	a+0=0+a=a	a.1=1.a=a

5.Inverse	∀a∈F	a+(-a)=0	a-1.a=0
6.Distributive	∀a,b,c∈F	a.(b+c)=a.b+a.c

Order:
Let  F be a field and let A,B∈F(here A and B are sets) then we define
                  A+B={x+y∈F : x ∈A and y∈B}

- A={-x : x∈A}
nA={nA: x ∈A} n∈N

In simple word :- A Field F is said to be ordered if there exist a subset P ⊂ F
such that
             (i)  P⋂(-P)=Φ
          (ii) P⋃{0}⋃(-P)=F
          (iii) a,b∈P  ⇒ a+b∈P and a.b∈P

[ i.e;P is a Field]

Bounded and Unbounded Set:
a set is called bounded if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded.

Upper Bound and Lower Bound of a Set:

A set M of real numbers is called bounded from above if there exists some real number k(not necessarily in M) such that k ≥ s for all s in M. The number k is called an upper bound of M. The terms bounded from below and lower bound are similarly defined.

A set S is bounded if it has both upper and lower bounds. Therefore, a set of real numbers is bounded if it is contained in a finite interval.

[The 'Greatest Lower Bound' is called 'infimum ' and 'Least Upper Bound ' is called 'Supremum' .]

The Completeness Property:

The Completeness Property of The Real Numbers: Every nonempty subset S of the real numbers that is bounded above has a supremum in R.

The Completeness Property is also often called the "Least Upper Bound Property".

The Archimedean Property:

Archimedean property which tells us that for any real number x there exists a natural number n that is greater than or equal to x. 

    [Simply;  ∀ x∈R there exist an element 

n∈N such that  x ≤ n .]

Proof:(By Method of Contradiction)
Letus assume x,y€ R
And for all n€N ,nx<y (contradictory statements)
Let A={nx: x ∈A} n∈Nand y is the upper bound.
By Completeness axiom,there exist q = lub(A).There exist mx€N such that q-x < mx
So q < x(m+1)
Which is a contradiction.

The Density of the Rational/Irrational Numbers:

Density: A subset S of R is called dense in R if between any two real number there must be a element of S.

Let x,y∈R be any two real numbers where x<y. Then there exists a rational number r∈Qsuch that x<r<y.

Similarly ,Let x,y∈R be any two real numbers where x<y. Then there exists a irrational number q∈R/Q such that x<q<y.

Dedekind form of Completeness Property:

😊

We will discuss from here in next chapter.