Archimedean principle of real numbers.pptx

Archimedean principle of real numbers

1. Archimedean principle of real numbers By :- Harsh Raj 20218211 Lakshya Sisodia 20218212 1

2. What does the Archimedean principle of real number state? The Archimedean principle states that any two distances are commensurable. We can find a finite multiple of the smaller distance that will exceed the larger. This specifically rules out the possibility of infinitesimal distances that are so small that no matter how many of them we take—as long as it is a finite number—we can never get enough to equal or exceed any finite length. IN is not bounded above in IR. This essentially means that there are no infinite elements in the real line. 2

3. IN is not bounded above in IR We can prove this by contradiction method… 3

4. We assume that N subset R is bounded above. Then there has to be a least upper bound of N. Let that upper bound be s. Now if n ∈ N, n+1 also ∈ N. Because n ≤ s, n+1 also ≤ s. That leaves us with n ≤ s-1 Now here s-1 also appears to be the least upper bound because not only is n ≤ s but also s-1. But by the definition of Least Upper Bound, for any positive number ε , however small, there exists a y ∈ N such that y > N- ε . Thus, in our case there must be a number which is definitely greater than s-1. THEREFORE, IN IS NOT BOUNDED ABOVE IN IR. 4

5. One very important result on the basis of Archimedean principle that asserts the finite nature of elements on a real line This result tells us that no infinitely small numbers on the real line exist. No matter how small ϵ gets, we will always find an n in the form of 1/n which is lesser than our given ϵ , where n ∈ IN. 5

6. This result comes handy to confirm and verify the limits to which sequences converge Convergence of a sequence to a limit is defined as A seq ( Xn ) of real no. converges to a limit X ∈ IR often written as Xn  X as n ∞ . If for every ϵ > 0, there exists N ∈ IN such that | Xn -X|< ϵ for all n>N. 6

7. Let us understand this with the help of an example Let’s verify that {1+(-1/2)^n} converges to limit 1. 7

8. We take an n ≥ N 8

9. ∴ And we know that ∴ Thus, we have verified that 1/2^n is indeed less than ϵ , thus it is proved that the sequence does converge to the given limit. And the Archimedean principle has played a pivotal role in this verification. 9