I understand that I probably asked a question that the users of this site would view as elementary, but I have only just dipped my feet into the waters of proof solving. Can somebody please tell me if my proof is valid?
Let there be two rational numbers $m/n$ and $p/q$ where $m$,$n$,$p$, and $q$ are integers, and $nq$ is not equal to zero.
$(m/n)<(p/q)$
$(p/q)-(m/n)=((pn-mq)/nq))$
The difference between the two numbers is a rational number. Two multiplied rationals has a rational product, and so if we multiply the (rational) difference by a rational number $>0$ and $<1$ and add that product to $(m/n)$ we have a rational number in between the two rationals.
Best Answer
Your proof is correct. A similar approach is just taking the average of the two rational numbers (which is again a rational number).
EDIT: As pointed out, the average is given by taking the rational number to be $\frac12$ to multiply the difference (using your method).