If a & b are any positive real numbers, integer n exists st na > b.

Proof:

Rational positive numbers r_{a}and r_{b}exist st r_{a}< a and r_{b}> b by rational cut definition of real numbers.

Then n > r_{b}/r_{a}= N_{b}/N_{a}→ na > b.*

N_{b}/N_{a}= (1/N_{a})N_{b}<= N_{b}< N_{b}+1 (Taylor)

n = N_{b}+1

*na > nr_{a}, & n > r_{b}/r_{a}→ nr_{a}> r_{b}> b, → na > b.