I'm trying to prove this statement:- "Let $x$ be a positive rational number. There are infinitely many positive rational numbers less than $x$."
This is my attempt of proving it:-
Assume that $x=p/q$ is the smallest positive rational number.
Consider $p/q – 1$
$= (p-q)/q$
Case I: $p$ and $q$ are both positive
Then, $p-q<p$
And hence, $(p-q)/q < p/q$
Since $p$ and $q$ are integers, $(p-q)$ is also an integer. Thus, $(p-q)/q$ is a rational number smaller than $p/q$. Therefore, our assumption is wrong, and there always exists a rational number smaller than any given rational number $x$.
Case II: $p$ and $q$ are both negative
Then, let $p/q = -s/-t$, where $s$ and $t$ are both positive integers.
Then, $-s-(-t)>-s \implies (-s+t)/-t < -s/-t \implies (p-q)/q <p/q$
Since $p$ and $q$ are integers, $(p-q)$ is also an integer. Thus, $(p-q)/q$ is a rational number smaller than $p/q$. Therefore, our assumption is wrong, and there always exists a rational number smaller than any given rational number $x$.
Q.E.D
Is my proof correct? And there are a couple of questions that I've been pondering over:-
1) How do I justify the subtraction of $1$ from $p/q$? I mean, I assumed that $p/q$ is the smallest rational number, so how do I even know if this operation is valid?
2) I proved that there always exists a smaller rational number given any positive rational number. But how do I prove that there's always a smaller positive rational number?
3) Also, I don't seem to have proved that there are infinitely many smaller rational numbers than $x$. If I use a general integer $k$ instead of $1$, this would be taken care of, right? But then again, how do I justify this subtraction?
I'd be really grateful, if someone could help me with this! Thanks!
Best Answer
Your proof does not work. Indeed, subtracting $1$ from $\frac p q$ will give you a rational number, but it will be negative by assumption, so this doesn't help you (since it doesn't give you a contradiction).
A simpler approach: Explicitly state what the infinitely-many positive rationals less than $x$ are.
Hint: If $y$ is a positive rational, what can you say about $\frac{y}2$? About $\frac{y}4$? $\frac{y}8$? ...