3 Sequences videos

In this post we will learn about sequences of real numbers. Sequence in itself is a function with domain as the set of Natural numbers and the co-domain as the set of real numbers.

The first video defines sequences using the epsilon- neighborhood explanation method. It will explain us the existence of a limit of a sequence.

In the second video we shall learn about some examples of different type of sequences and finally in the last video we will prove that any real sequence can tend towards only one limit say L, and that L has to be unique. Continue reading →

4 Intervals of real number

In this post we will discuss about all types of Interval of real numbers. What are Open intervals? What are closed intervals? What are half open – half closed intervals?
In the second video we shall be characterizing the intervals of real numbers by asserting that any subset of real numbers containing atleast two points and following the order property can make an interval of those two points.
In the third video we will understand the property of making nested intervals. What do we mean by the nested sequence of intervals.
In the fourth video we will try to figure out a sufficient condition for a nested sequence of intervals to have a common point.
Finally we will discuss the countability of some important sets like Q and R. Continue reading →

4 Convergence of sequences videos

Now that we know what are sequences and what do we mean by the limit of any sequence we shall learn different ways to express the same. Hence in the first video we will learn about various definitions for convergence of a sequence and understand how they are equivalent.

In the second video we will use the epsilon definition in order to find the limit of some sequences and in the third video we will try to understand an m tail sequence out of any sequence and how if a sequence converges the limits should be the same for m-tail and original sequence.

In the last video of this post we shall understand the sandwich theorem for sequences just like we have learnt for limits of any function. Continue reading →

4 Supremum Properties for a set

There are four main properties of Supremum that we should be aware of. In the first video we will see how the supremum of a set a+S is equal to the sum of a and supremum(S).

In the second video we will lear how to relate supremum and infimum of two distinct related sets. We will check when and in what conditions will SupA < Inf B for any given sets A and B.

In the third video we will try and understand what is the relation between the Supremum of any two bounded functions . if f(x)<g(x) for all x, then what is the relationship between the supremum of f and g.

Finally, in the fourth property we twist the previous property and check the relationship between these two functions given the condition that f(x)<g(y) for all x and y. Continue reading →

3 Bounded Function videos

In the following videos we will try to evaluate what are bounded functions. What is the meaning of a bounded function? How is a bounded function different from a bounded set?

We shall be looking at some examples of such functions, explaining the concept of boundedness.

Finally in the last video we will look at an example of a function f(x) = 2- 1/x, which is bounded above but not bounded below. Continue reading →

3 more Supremum and Infimum concepts

Having discussed the least upper bound of a bounded above set, in the first video today we will discuss the concept of the greatest lower bound of a bounded below set.
In the second video we will show with examples that we might have many upper bounds and lower bounds but the supremum and infimum are unique always.
Finally we will work out the formal characterization of the supremum for any set. This is another way of stating the definition of supremum, which is that no number smaller than the supremum can be an upper bound of the given set. Continue reading →

1 Concept on usability of Derivative: Speed

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In this video we will relate the concept of speed with derivative.

We will cover following issues in these videos-

Physical speed interpretation of a derivative Continue reading →

2 Videos deriving the derivative of x^n

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In the following videos we will discuss the differentiation process of x^n in 2 different ways.

We will cover following issues in these videos :

Extension to derivative x^n
Derivative of x^n Continue reading →

3 limits ,continuity & discontinuities concepts

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The following videos will discuss the concept of limit, continuity and discontinuities of function at a given point.

We will cover following issues in these videos-

Defining Limit of a Function
Defining Continuity
Discontinuities

Continue reading →

3 Neighbourhoods and bounded sets concept

The next 3 videos will throw light upon the concept of neighbourhoods and upperbounds and lowerbounds.

The first video will define the concept of a neighbourhood of a set and interpret it on a real line. The second video will then make us understand what the bounds of a set are. The concept of upper bounds and lower bounds will in turn let us find the greatest lower bound and the lowest upper bound for any set.
Finally in the third video we will talk about the lowest / least upper bound of a set which is called the supremum. Continue reading →

4 More properties of Absolute value

Next 4 videos are a continuation of the last post, in here we will learn some more properties related to the absolute value of real numbers. In the first video we will study the concept of Triangular Inequality.

Property 7 and 8 relate the inequality behavior when two real numbers are added or subtracted.
Namely, ||a|-|b||< |a-b| and |a-b|< |a|+|b|. we will check how this happens.
In the last video we will prove why the property that says |x-1|<|x| holds true for all real numbers x. Continue reading →

4 Absolute value of Real numbers concepts and Properties

In the next few videos we will work in detail on the concept of absolute value or modulus of real numbers. In the first video we will define it with examples and then further we will look at the properties related to modulus that are true for all real numbers.

In the second video we will understand the property that says |a.b|=|a||b| and that this holds true for all real numbers.

In the third video we will proof why |a|^2 is equal to a^2 in all circumstances, even when a is negative. In the fourth video we will proof two properties, property 3 and 4.

Property 3 shows how we can interpret modulus in terms of an interval and in property 4 we will learn that the real number a lies between the positive and negative values of the absolute value of a. Continue reading →

3 Famous inequalities of Real Number

In the next 3 videos we will work around some basic and famous inequalities related to Real numbers. In the first video we will proof that if product of two real numbers is greater than 0 then either they both are positive or they both are negative.
In the second video we will try understanding that why is the arithmetic mean always greater than the geometric mean. For example why is AM of 4 and 9 (4+9)/2 = 6.5 > the GM of 4 and 9 which is 6.
Finally we will learn about the Bernoulli’s inequality and we will also prove it using Induction. Continue reading →

4 Properties of the Real numbers

4 videos on further properties of Real numbers apart from the axioms, infact they are deduced from the axioms only. First video is on the uniqueness of the neutral element of addition and multiplication operations.

The second video proves that multiplication of a real number with the number 0 is always 0. In the third video we will prove the uniqueness of inverses with respect to addition as well as multiplication. In the last video we will prove that if a.b=0 then either a=0 or b=0 for any real number a and b. Continue reading →

2 Real numbers and Field axioms concept

2 videos on the concept of Real numbers and sets around it. In the first video we will learn what do we mean by a set of numbers, certain notations related to sets, the set of no element that is phi. Then we will specify all the subsets of the set of Real numbers i.e Natural numbers, rational numbers, irrational numbers and integers and a comparison of all.
In the second video we will talk about the algebraic properties related to R. These include the field properties, the order properties, completeness. These axioms show that R is an ordered Field. Continue reading →

3 Order/ and related Properties of Real number

The next 3 videos build up the concept of Order structure of the real numbers. To understand the order properties we will first define the set of positive numbers and use it to prove the further properties.

In the first video we will look at the axioms related to the set of positive numbers and then prove the Trichotomy law of real numbers.

In the second video we will try to relate the algebraic properties of R to the order properties. Finally, in the last video explains the consequence of these properties, we will explain why 1>0 and that all natural numbers are greater than n. Continue reading →

3 Modulus ‘function’ facts we must know

Continuing with our objective of learning mathematical functions let us now see how to deal with a Modulus function.

The videos in this post will make you understand the meaning of a modulus function and how it makes every real number positive. We shall also learn about all the properties related to a modulus function involving real numbers. Continue reading →