The document discusses the Archimedean principle of real numbers. It states that the Archimedean principle rules out the possibility of infinitesimal distances that are so small that no finite amount can exceed any finite length. It also means that there are no infinite elements in the real number line. The document proves that the natural numbers are not bounded above in the real number line using contradiction. It explains that the Archimedean principle asserts the finite nature of elements on the real line and is useful for confirming the limits that sequences converge to.

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