Density of rational numbers in R

This theorem states that between any two real numbers we can always find a rational number.
Consider a Real number line, starting with 0 and having two points a and b on the number line
0--------------------a-------b-------

Our objective here is to find a rational number (lets say m/n) between the interval (a,b)
Lets divide this line into subsections, each of which is less than the width of interval (b-a). Using Archimedian property we can always find a natural number n such that 1/n < (b-a)

0-1/n-2/n--(m-1)/n-a-(m/n)---b---

Thus we see that smallest value of m which makes the ratio a > (m)/n is (m-1)/n < a . Now we have to make sure that ratio m/n is smaller than b. Substituting for the value of a in the equation 1/n < (b-a) we get
b > 1/n+(m-1)/n
b > m/n
hence proved