Prove by Archimedian property

Here is the problem:

Use the Archimedian property to prove that if a,b belongs to R (real number) with 0< a < b, then there is a rational number p/q with p and q both odd such that a<p/q<b. (Remark: Modify the proof that rationals are dense.)

Here is my work:

By the Archimedian property, there exists an odd q belongs to R such that 1 < qa implies that 1/q < a. By the Archimedian property agian, there exists an odd p belongs to R such that a<p/q<b.

Is it right???

Can you show that if then there is an integer such that
If so we know there us an integer such that So is between them.

Thank you ~ problem solved