Archimedian property

 APPLICATIONS    OF  THE SUPREMUM PROPERTY

Theorem : (Archimedian property) :

If x > 0, then for any y € R, there exists n €N

such that nx > Y'

Proof :

 Case (i) : 

           When y ≤ 0, the theorem is obvious.

       Case (ii) :

When y > O,

Let us assume that the theorem be false.

Therefore nx ≤ y for all n € N.

Hence the set S = {nx | n € N} is the

non-empty set bounded above, since nx ≤y for all n € N.

Let u = sup of S, then nx ≤u for n € N.

                      (n+1)x ≤ u for all n€N

 

                    n x≤u—x for all n € N

                      

Thus u — x is also an upper bound of S, but u — x < u, which contradicts that u = sup S. Hence nx > y for some n € N.

---- Thus proved -----