Thread: Q is dense in R

Use the density of Q in R to prove the Archimedian property of R.
I know Q is dense in R refers to theres a "p" that exists in Q that is x>p>y where x and y exist in R, but how can i use this to prove the archimedian property of R?

Suppose that we need to find a natural number n greater than a given real number x. Choose any y > x, e.g., y = x + 1. Then by density there is a rational number y > p / q > x where p and q are integers and without loss of generality q >= 1. Multiply by q to get p > xq >= x.