Let x and y be real numbers with x<y and write an inequality involving a rational
number p/q capturing what we need to prove. Multiply everything in your inequality by q,
then explain why this means you want q to be large enough so that q(y-x)>1 . Explain
how you can rewrite this inequality and use the Archimedean property to find such a q.
The Attempt at a Solution
So, this is a question on a worksheet our teacher gave us to go along with the theorem in the book. Here is what I did so far:
x < p/q < y
Then multiply both sides by q as the question states:
qx < p < yq
0 < p-qx < yq-qx
0 < p-qx < q(y-x)
I am having trouble seeing why q(y-x)>1. It is obviously great than zero as the inequality states, but can someone help me see why it has to be >1??