Posted by Anjum Shekh at April 14, 2022

**Upper bound** of set **S = { x _{1}, x_{2,} x_{3} ,x_{4},………..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={ x_{1}, x_{2} ,x_{3,} x_{4},…….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 m_{1}, m_{2}, m_{3}…….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={x_{0=}a, x_{1}, x_{2}, x_{3}………x=b}

Such that x_{0}<x_{1}< x_{2} x_{3} x_{4}……. <x_{n-1<}x_{n }is called partition of [a, b].

**Refinement of Partition:-**

Let P_{1 }and P_{2 }are any two partition of interval [a, b] Then P_{2 }is said to be refinement of P_{1 }if P_{1 }⊆_{ }P_{2}

**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={x**_{0=}**a, x**_{1}**, x**_{2}**, x**_{3}**………x=b}** is any partition of [a, b] Then [x_{0}, x_{1}], [x_{1}, x_{2}]

[x_{2}, x_{3}]… [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={x_{0=}a, x_{1}, x_{2}, x_{3}………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={x_{0=}a, x_{1},x_{2}, x_{3}………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)

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={x**_{0=}**a, x**_{1}**,x**_{2}**, x**_{3}**………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 M_{1}, M_{2}, 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 m_{1}, m_{2}, 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/n**^{2}**>**

**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 ∃ n_{0} ∈ N

Such that **|a**_{n-l }**|<∈ ∀ n ≥ n**_{0}

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 ∃ n_{0} ∈ N such that **| a**_{n}**-a**_{m }**|<∈ ∀ n, m≥ n**_{0}

**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.

- If d(x, y) ≥ 0
- d(x, y) = 0 iff x=y
- d(x, y) =d(y, x)
- 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.

- If d(x, y) ≥ 0
- d(x, y) =d(y, x)
- d(x, y) ≤ d(x, z) + d(z, y) ∀ x, y, z ∈ X

**Open Sphere:-**

Let <x, d> is a metric space and x_{0 }∈ X is a fixed element if (r) is any non-negative real number then the set **S(x**_{0}**, r) = {x ∈ X: d(x, x**_{0 }**)<r}** is called open sphere OR open ball the point x_{0 }is called center and r is called radius of sphere it is generally denote by S_{r }(x_{0 }) or S(x_{0,}r) or B_{r }(x_{0 }) 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 x_{0 }∈ X is a fixed element if r is any non-negative real number then the set **S[x**_{0}**, r] = {x ∈ X: d(x, x**_{0 }**)≤r}** is called close sphere OR open ball the point x_{0 }is called center and r is called radius of sphere it is generally denote by S_{r }(x_{0 }) or S(x_{0,}r) or B_{r }(x_{0 }) or B(x_{0, }r).

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