prove there is no smallest positive rational number How would I prove there is no smallest positive rational number?
what is the best method to prove this statement?
 A: Assume that $q$ is the smallest positive rational number. But then clearly $\frac{q}{2}$ is also rational and 
$$0 < \frac{q}{2} < q.$$
A: Let $\epsilon > 0$ be given. There exists $n \in \mathbb{N}$ such that $n > \frac{1}{\epsilon}$. Then $0 < \frac{1}{n} < \epsilon$. Hence there can exist no positive lower bound on positive rational numbers.
A: Let $x$ be the smallest positive rational number. Then $y = \frac{x}{2} \in \mathbb{Q}$ but $0 < y < x$. Contradiction.
A: A proof by contradiction is rather simple:
Assume that the smallest rational number exists and is of the form: $a/b$
Then note that we can define $a/(b+1)$, which is rational as it is the quotient of 2 integers, and is strictly smaller than $a/b$ as its divisor is greater. Hence we contradict our initial statements that $a/b$ is the smallest possible rational number, so by this contradiction, we know that there is no smallest rational number. 
So since we can always define a smaller rational number than the one we have, there can be no smallest rational number.
A: Since You're proving by contradiction, the statement "There is no smallest rational number." becomes "There is a smallest rational number."
We can prove this by saying:
Let r be any rational number. Since r is a rational number we know that r/2 is also rational. Because r/2 is rational, we can assume r/2 < r, therefore there is no smallest rational number.
