Cauchy sequence

A Cauchy sequence is a sequence whose terms become very close to each other as the sequence go further. Cauchy sequence is helps in study of two topics
1. In real number
2. In metric space

Cauchy sequence

In real number

Definition:- A sequence {xₙ} is said to be Cauchy sequence if given ε>0, however small, ∃ a positive integer k(depending upon ε) such that

                  

Another definition:- A sequence {xₙ} is said to be Cauchy sequence if given ε>0, however small, ∃ a positive integer m(depending upon ε) such that

                

In given figure, we have taken n as subscript on x-axis and xₙ on y-axis. when n,m≥k, take  very close to 1 then  is very small number. This implies that the sequence is Cauchy sequence and converge to l.

Article 1:- Prove that a Cauchy sequence is bounded.
proof:  Let {xₙ} be Cauchy sequence.
        given ε>0, there exist a positive integer p such that
                
In particular, 

               
Now     

                   
                   
              
Let 
      | xₙ |≤M   ∀ n
⇒   {xₙ} is bounded

Article 2:- Prove that a convergent sequence is always a Cauchy sequence.
Proof: Let the sequence {xₙ} converge to l
          given ε>0, however small, ∃ k∈ℕ s.t
                
Let m≥k be a natural number.
                
Now    
                            
                            
                            

               
             {xₙ} is a Cauchy sequence.

From artical2 it is clear that when ever a sequence is Cauchy sequence, it is convergent to.

In Metric space

Definition:- A sequence {pₙ} in a metric space X is said to be a Cauchy sequence if for every ε>0 there is an integer N such that   if n≥N and m≥N.
   
Definition:- Let E be a non empty subset of metric space X, and let S be the set of all real number of the form d(p,q), with p∈E and q∈E. The sup of S is called the diameter of E.
     If {pₙ} is a sequence in X and if Eₙ consist of the point   it is clear from the two preceding definition that {pₙ} is a Cauchy sequence if and only if

                           

Definition:- A metric space in which every Cauchy sequence converges is said to be complete.
         Thus every Cauchy sequence in E is a Cauchy sequence in X, hence it converge to some p∈X, and actually p∈E since E is closed. An example of metric space which is not complete is the space of all rational number, with d(x,y)=| x-y |.
         It shows that convergent sequence are bounded, but that bounded sequence in Rⁿ need not converge. However, there is one important case in which convergence is equivalent to boundedness; this happen for monotonic sequence in R¹.

Convergence

The convergence sequence is the value, sequence approaches as the number of terms goes to infinity. Not every sequence is convergent. The sequence whose nth term approaches are called convergent while those that don't are called divergent. To understand what convergent is you first have to understand what sequence is?.

Sequence

Definition:- If ℕ is a set of natural number and X any set, then a function f :ℕ→X is called sequence.
If X=ℝ or a subset of ℝ, then f is called a real sequence and if X=ℂ or a set of ℂ, then f is called a complex sequence.

Let a :ℕ→ℝ be a sequence. The image of n∈ℕ, instead of being denoted by a(n) is generally denoted by aₙ thus a₁a₂a₃a₄....... are the real number associated to 1 2 3 4.....by this mapping aₙ is called the general term or the nth term of the sequence.

Examples of sequence

1. The sequence {aₙ} where aₙ=n is {1,2,3,...n,....}
2. The sequence {aₙ} where   is {}
3.The sequence {aₙ} where aₙ=(-1)ⁿ is {-1,1,-1,1,-1,.....}

Convergence sequence

A sequence {aₙ} is said to converge to a limit l, if given ε>0 however small there exist a positive integer m(depending upon ε) such that
        |aₙ-l|<ε   ∀ n≥m
l is called a limit of sequence {aₙ} and we write it as
                   
or aₙ→l as n→∞  or simply aₙ→l

We know that |aₙ-l|<ε  ⇒  aₙ∈(l-ε,l+ε). Now a neighbourhood V of l always contain (l-ε,l+ε). Thus we gives another definition of convergent sequence.
A sequence {aₙ} is said to converge t a limit l if given a neighbourhood V of l, there exist aₙ integers m such that aₙ∈V ∀n≥m

Meaning:- At a point when n become greater than m(n≥m). Then every term of aₙ reaching to a particular value. Then we call it as {aₙ} converges to l.

In the figure, aₙ is taken on y-axis and its subscript on x-axis. It show that if all the terms after m lies between (l-ε,l+ε) then sequence {aₙ} converges to l.

For example the sequence  converge to 0

Infimum and Supremum

The infimum and supremum are concepts in mathematical analysis. It helps in study of two subjects. It describe the minimum and maximum of finite set. The limit of the infimum and supremum used in some convergence sequence and in particular in computation of domain of convergence of power series.

In set theory

To understand the concept of infimum(glb) and supremum(lub), you first have to understand what is upperbound and lower bound.

Upper bound:- Let S be a non-empty subset of real number.
If there exist a real number K such that x≤k ∀ x∈S
then k is said to be an upper bound  of S the set S is said to be bounded above or bounded to the right.

Meaning:- It means that if x∈R then there always exist a number which is greater than x.
Example:- If x=5 then take k=6,7,8,9..... they are all upper bound of x and are greater then 5.

Lower bound. If there exist a real number l such that l≤x ∀ x∈S
Then l is said to be lower bound of S. The set S is said to be bounded below or bounded to the left if it has a lower bound.

Meaning:- It means that if x∈R then there exist a number which is lower then x.
Example:- If x=5 then take l=4,3,2,1..... they are all lower bound of x and are lower then 5.

Supremum (least upper bound)

Let S be a non-empty subset of R which is bounded above. Then there exist a real number u which is the smallest of all upper bound of S i.e, for other upper bound u´ of S we have u≤u´.
This number u is called the least upper bound of S or supremum of S.

Meaning:- Least upper bound of a set S bounded above is unique. For, if u and u´ are two least upper bound of S, then by definition, u≤u´ and u´≤u which implies that u=u´.

Infimum (greatest lower bound)

Let S be a non-empty subset of R which bounded below. Then there exist a real number l which is the greatest of all the lower bound of S i.e., for any other lower bound l´ of S we have l´≤ l
This number l is called the greatest lower bound of S or infimum of S.

Example. if x∈[-3,10]
                     -3≤x≤10
                      glb=-3,   lub=10

Sequence

Definition:- If N is a set of natural number and X is any set, then a function f: N→X is called a sequence.

Let a: N→R be a sequence. The image of n∈N is denoted by aₙ. thus a₁,a₂,a₃....... are the real number associated to 1,2,3....... by this mapping aₙ is called the general term or the n term of the sequence.

bounded and unbounded sequence

bounded above=A sequence {aₙ} is said to be bounded above if there exist a real number k such that aₙ≤k  ∀ n∈N
k is called an upper bound of the sequence {aₙ}.

bounded below=A sequence {aₙ} is said to be bounded below if there exist a real number h such that h≤aₙ n∈N. h is called a lower bound of the sequence {aₙ}.

bounded sequence=A sequence {aₙ} is said to be bounded if it is a bounded above as well as bounded below i.e., if there exist two real number h and k such that
h≤aₙ≤k ∀ n∈N.

supremum(lub)
1. aₙ≤u   ∀ n∈N.
2. Given ε>0 however small,∃ at least one positive integer m such that a >u-ε.

infimum(glb)
1. aₙ≥l   ∀n∈N.
2. Given ε>0 however small, ∃ at least one positive integer m such that <l+ε.

The definition of limit

In calculus, the ε-δ definition of a limit is a finest form of limit of a function. The definition states that a limit L of a function at a point α exist if no matter how α is approached, the value returned by the function will always approach L.

Formal definition of Epsilon Delta limit

Definition of limit. A function f is said to have a limit l as x→a written as
                      
given ε>0 however small,∃a positive real number δ(ε) such that

Meaning of x→a(x approaches a)

x→a means  but | x-a | is very small. so x→a means there exist a positive number δ>0, however small,such that 0<| x-a |<δ
or   x∈(a-δ,a+δ) and 
or   x∈(a-δ,a)∪(a,a+δ)
if x∈(a-δ,a) only, then we say that x tends to a from the left and we write it asx→a- or x→a-0. similarly when x∈(a,a+δ) only, then we say that x tends to a from right. We write it as x→a+ or x→a+0.

left limit: A function f is said to have a left limit l as x→a-, written as , if given ε>0, however small, there exist a positive real number δ(ε) such that
| f(x)- l |<ε for a-δ<x<a 

Note. δ(ε) means δ depends upon ε,

Right limit. A function f is said to have a right limit l as x→a+, written as , if given ε>0, however small, there exist a positive real number δ(ε) such that
| f(x)- l |<ε for a<x<a+δ.

For example 1 :- In the graph for function f(x) below if Jame and tom the value ε, then tom gives him the number δ such that for any a in the open interval (a-δ,a+δ), the value of f(x) lies in the interval (l-ε,l+ε). In this example, as Jame make ε smaller δ satisfying this property, which show that the limit exist.

Example 2 :- Now think of the opponent ε challenge as a vertical target around L and your δ response is a horizontal shooting range around a. What does it means for your δ to be successful? It means that whenever you stand in your shooting range (except for standing at the point a itself), and you shoot, you make it into ε target, as in the picture given.

            The small an ε your opponent choose the smaller the target, and the harder your job is. You may have to pick a correspondingly smaller δ. so this implies that no matter what ε is you can always find some δ response that work, you've won the game and prove that

                

Question 1:- By use of definition of limit, show that 

Solution:- Let f(x)=4x-5,   l=3

         | f(x)-l |=| (4x-5)-3 |=| 4x-8 |=4| x-2 |

                 Let ε>0, however small, be given 

              now | f(x)-l |<ε     whenever 4| x-2 |<ε

                i.e.,   =δ(say)

given ε>0 however small, we can find a positive number δ(ε) such that

              | f(x)-l |<ε   for    0<| x-2 |<δ

              

How To Memorise Formulas

You spend the long night to learn mathematical formula, but at the very next day. When you try to recall it, it just feel like vanish from your brain.

In maths, forgetting formula is normal, you can't learn all formula's by cramming. So you have to use right memorising techniques for learning maths formula's. The main reason why you can't memorise the formula are

• Rare usage:- Those formula include which are rarely used in the chapter or book.

• Wrong memorising technique:- most of the time student use wrong memorising technique for learning the formula. So in exam we forgot those formula's.

The use of right memorising technique help to retain the formula for longer period. Below there are 5 techniques that helps you to memorise formula.

5 techniques to memorise formula's

1. Mnemonics 

Mnemonic is any learning technique that aids information retention in the human memory. As knuckle mnemonic is used to remember the number of days in a month.

Example 1:- You must have learned the formula of sine cousin and tangent.

               Some      curly      turn   

               People    brown   pink

               Have       hair       blueu

This means

Where p=perpendicular

             H=hypotenuse
             B=base

Example 2:-we can also make definition of logarithm easier

 

If we take the alphabet x,y,z this formula is hard to learn. But when we use the concept of answer,base,power, we can easily remember this. Through the use of mnemonics some formula's can be easily learned.

2. Pattern

In pattern you have to find relation between two different formula and try to relate each other.

Example:-

                     

                 
In this formula as we know the derivative of  , therefore we can able to find the integration of 

3. Create a list of maths formula's

Make a list of all formulas whenever you newly learn it. List will be similar to dictionary. Make sure to write each and every formula in a list. If you forgot any formula, just revise from the list. The list is extremely  beneficial at the time of exams.

4. Repetitive usage

The formula which we repetitively use, we always remember that formula.

For example:- 

You always remember this formula. This formula is Repetitively used by everyone.

5. Understand the formula

Whenever you learned a new formula, you try many question related to that. To properly understand the formula, you have to relate the formula to real world.

For example:- you know that speed is equal to distance upon time. When you visualise this formula with the help of a car you easily remember this formula.