Real Analysis

Posted by Anjum Shekh at April 14, 2022

Upper bound of set  S = { x₁, x_2, x₃ ,x₄,………..x_n } is  non- empty  set  then a finite real number m is said to be an Upper bound of S.

If x_i <= m ∀i=1, 2, 3, 4……… also all the real number which are greater than M are also upper bound of set S.

S = {-1,-2,-3,-4……….} x_i ≤ (-1)=M

Lower bound:- 

Lower bound of set S={ x₁, x₂ ,x_3, x₄,…….x_n} is any non-empty set Then a finite real number m is said to be a lower bound of S is if x_i ≥m ∀i=1,2,3,4,……… also all the real number Which are less than m are also lower bound of set S.

Bounded Above Set:-

A set S is said to be bounded above set if it have an upper bound of set.

Example. {-1,-2,-3,-4………}

Bounded Below Set:-

A set S is said to be bounded below set if it have a Lower bound of set.

Example. {1, 2, 3, 4………}

Bounded Set:-

A Set S is said to be bounded set if have both bounded above and bounded below set.

OR

A Set is said to be bounded set if it’s have both upper and lower bound of set S.

Exam. S ={-1,1}

Least Upper bound (LUB) OR Suprimum:-

Let S be a bounded above set and m1,m2,m3,m4………… are all upper bound of set S then the least member among these Upper bound is called Least Upper bound(LUB) or Suprimum of set S.

S= {-1,-2,-3……..}

M= {-1, 0, 1, 2, 3)

Greatest Lower Bound (GLB) OR Infimum:-

Let S be a bounded below set and m₁, m₂, m₃…….Are all lower bound of set S Then the least member among these lower bound is called great lower bound (GLB) or Infimum of set S.

Exam… S= {1, 2, 3, 4…)

Interval:-

Let a and b are any two real number such that a<b Then the interval are defined as.

1. [a, b]={x :a ≤x ≤b} it is closed

2. (a, b)={x: a ≤x ≤b} it is Open

3. (a, b] ={x: a ≤x ≤b} it is semi-open from Left or semi closed from right.

3. [a, b) ={x: a ≤x n ≤b} it is semi-closed from Left or semi open from right.

Partition of an Interval:-

Let I= [a, b] is any closed interval Then a finite set P={x₀₌a, x₁, x₂, x₃………x=b}

Such that x₀<x₁< x₂ x₃ x₄……. <x_n-1<x_n is called partition of [a, b].

Refinement of Partition:-

Let P₁ and P₂ are any two partition of interval [a, b] Then  P₂ is said to be refinement of P₁ if P₁  ⊆  P₂

Range set /Image Set:-

Let f:[a, b]→R is a function Then the range set of f( or image set of f) is defined as.

Range (f) = {f(x):∀x∈ [a, b]}

Bounded function:-

A function f:[a, b]→R is said to be a bounded function if ∃ two finite real number m& M such that m≤f(x)≤M

Sub-Interval:-

Let [a, b] is any closed interval and P={x₀₌a, x₁, x₂, x₃………x=b} is any partition of [a, b] Then [x₀, x₁], [x₁, x₂]

[x₂, x₃]… [x_k-1, x_k], [x_k+1, x_k+2]…………., [x_n-1, x_n], 

Are sub-interval of [a, b] w.r.t Partition (p).

Note. If P={x₀₌a, x₁, x₂, x₃………x=b} is any partition of [a, b] Then sub-interval of [a, b] w.r.t Partition is generally by ..[Xk-1, x_k] ∀K=1, 2, 3, 4 …n

Note: – let f:[a,b]→R is a bonded function V and P={x₀₌a, x₁,x₂, x₃………x=b} is any partition of [a, b] and [x_k-1, x_k],

K=1, 2, 3, 4, 5……..n are sub-interval of [a, b] w.r.t Partition P.

M_k= inf {f(x): x∈ [x_k-1, x_k]}

m_{k =} Sup {f(x): x∈ [x_k-1, x_k]}

m = inf {f(x): x∈ [a, b],}

M = Sup {f(x): x∈ [a, b],}

Note: – m≤ m_k≤ M_k≤ M

 Length of Sub-interval:-

Let f (a, b)→R is a bounded function and P={x₀₌a, x₁,x₂, x₃………x=b} any Partition of [a, b] and [x_k-1, x_k] ∀

K=1, 2, 3, 4, 5……..n are Sub-interval of [a, b] w.r.t Partition P. Then the length of Sub-interval [x_k-1, x_k] is denoted by △k and is defined as. △k= [x_k-1, x_k]

Sequence of Real Number:-

Sequence:- 

Let f: N→R is any function Then <f (1), f (2), f(3), …….> is called a sequence in R. if <f(n)> = <a_n>

Here a_{n =} n^th term of the sequence.

Example: Let f: N→R

Such that f (n) =1/n

Then <a_n>= <1, 1/2, 1/3, 1/4 ……….> 

Upper Bound of a Sequence:-

Let <a_n> is any sequence Then a finite real number M is said to be an Upper bound of Sequence if a_n ≤ M ∀ n=1,2, 3, 4, 5, …… also all the real number greater than M are Upper bound of Sequence.

Lower Bound of a Sequence:-

Let <a_n> is any sequence Then a finite real number m is said to be an Lower bound of Sequence if a_n ≥ m ∀ n=1,2, 3, 4, 5, …… also all the real number less than m are Lower bound of Sequence.

Bound Below Sequence:-

A Sequence <a_n> is said to be bounded below Sequence if it have Lower bound of Sequence.

Bound Above Sequence:-

A Sequence <a_n> is said to be bounded above Sequence if it have Upper bound of Sequence.

Supremum (Least Upper Bound):-

Let a sequence <a_n> is bounded above and M₁, M₂, M3_, M_{4 …}……. Are all Upper bound of sequence <a_n> Then the least member among these upper bound is called least upper bound (L.U.B) or Suprimum of <a_n>.

Infimum (Greatest Lower Bound):-

Let a sequence <a_n> is bounded above and m₁, m₂, m3_, m_{4 …}……. Are all Lower bound of sequence <a_n> Then the greatest member among these lower bound is called Greatest Lower bound (G.L.B) or Infimum of <a_n>.

Note: –  A Sequence <a_n> will be bounded sequence iff ∃ a finite real number k such that | a_n |≤k that is k ≤ a_n ≤ k.

Range Set of Sequence:-

Let <a_n> is any sequence then the set of all district element of sequence <a_n> is called range set of sequence <a_n>  it is denoted by range <a_n>.

Exam… <a_n>= <1, 1/2, 1/4 >

Constant Sequence:-

A sequence <a_n> is said to be constant sequence if it is Range set is singular set.

Exam… {1, 1, 1, 1, 1,} 

Note: – 1- A sequence have infinite number of term.

2. The range set of sequence can have finite and infinite number of element.

Monotonic Non- Decreasing Sequence:-

A sequence <a_n> is said to be monotonic Non decreasing sequence if a_n ≤ a_n+1 ∀ n∈ N

Exam.. <a> = <-1/n>

Monotonic Non- Increasing Sequence:-

A sequence <a_n> is said to be monotonic Non Increasing sequence if a_n ≥ a_n+1 ∀ n ∈N

Exam.. <a> = <1/n>, <a> = <1/n²>

Monotonic Sequence:

A sequence <a_n> is said to be monotonic sequence if it is either monotonic none increasing or monotonic none decreasing sequence.

Note:    A sequence <a_n> which is both monotonic non-increasing and monotonic non-decreasing sequence then it is called constant sequence.

 Convergent Sequence:-

A sequence <a_n> is said to convergent sequence to a finite real number l.

If ∀ ∈>0 ∃ n₀ ∈ N

Such that |a_n-l |<∈ ∀ n ≥ n₀

Where l is called limit of sequence.

Cauchy Sequence or Fundamental Sequence:-

A sequence <a_n> is said to be Cauchy sequence if ∀ ∈>0 then ∃ n₀ ∈ N such that | a_n-a_m |<∈ ∀ n, m≥ n₀

Divergent Sequence:-

A sequence <a_n> is said to be a divergent sequence if it is not convergent if it is not have a limit.

Exam. <a_n>= <n>

Oscillatory Sequence:-

 A sequence <a_n> is said to be oscillatory sequence it neither convergent nor divergent.

Exam… <a_n> = <(-1)ⁿ>

Metric:-

Let X is a non-empty set then a mapping d: X *X→R is said to metric on X.

1. If d(x, y) ≥ 0
2. d(x, y) = 0 iff x=y
3. d(x, y) =d(y, x)
4. d(x, y) ≤ d(x, z) + d(z, y) ∀ x, y, z ∈ X

Metric Space:-

Let X is a non-empty set and d: X* X→R is metric on then <x, d> is called metric space.

Usual Metric:-

Let R be a the set of real number then the map d: R*R→R defined by d(x, y)= |x-y| ∀ x, y ∈ R is a metric on R then it is called usual metric.

Pseudo metric:-

Let X is a non-empty set and d: X*X→R is a map which is satisfied following condition.

1. If d(x, y) ≥ 0
2. d(x, y) =d(y, x)
3. d(x, y) ≤ d(x, z) + d(z, y) ∀ x, y, z ∈ X

Open Sphere:-

 Let <x, d> is a metric space and x₀ ∈ X is a fixed element if (r) is any non-negative real number then the set S(x₀, r) = {x ∈ X: d(x, x₀ )<r} is called open sphere OR open ball the point x₀ is called center and r is called radius of sphere it is generally denote by S_r (x₀ ) or S(x_0,r) or B_r (x₀ ) or B(x_0, r).

 Note: – Open sphere or open ball or open disc or open cell are the same.

Close Sphere:-

 Let <x, d> is a metric space and x₀ ∈ X is a fixed element if r is any non-negative real number then the set S[x₀, r] = {x ∈ X: d(x, x₀ )≤r} is called close sphere OR open ball the point x₀ is called center and r is called radius of sphere it is generally denote by S_r (x₀ ) or S(x_0,r) or B_r (x₀ ) or B(x_0, r).

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