5 Archimedean Property of the real numbers and associated corollary

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The Archimedean property of R explains that for any real number x there always exists a natural number after it i.e. x<n. intuitively this property simply means that the set of natural numbers is unbounded because it is not bounded above and that what we will prove in the first video.

In the next video we will prove a corollary of this property which explains why the set of all numbers of the form 1/n will have its Infimum as 0.

In the third video we will see how the Archimedean property is used to prove why for any real number we can find a natural number such that 0< 1/n <x.

In the fourth video in this post gives us two more corollaries of the Archimedean property which tell us that any interval [n,n+1) would contain many real numbers, and also the existence of a square root.

In the final video we will discuss the Density theorem that comes through the Archimedean property and asserts the existence of many rational numbers between any two real numbers. 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 →

Bounded sets

Bounded sets 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 →