5 Archimedean Property of the real numbers and associated corollary

September 12th, 2012 BY vidhi (0 comments)

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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.