# Showing Density in R

1. Oct 5, 2011

### Punkyc7

if u>0 is any real number and x<y show there exist a rational number r such that x<ru<y. Hence {ru: r$\in$Q is dense in R.

I am not sure how to show that there exist a rational number. I was thinking this has something to do with the archimedian property. This is what I have tried

x<$\frac{(x+y)}{2}$<y

Define r= $\frac{(x+y)}{2}$(1/n) when u>=1 and
r=$\frac{(x+y)}{2}$(n) when u<1

where n is an element of the naturals

but I dont think that works? Any suggestions?