Prove that There is no smallest rational number greater than

Prove that: There is no smallest rational number greater than 0.

Solution

Answer:

To prove the given statement ,let us take a contradiction to the given statement. i.e., there exists a smallest positive rational number greater than zero, call it R. Since R is rational and R>0, we can write it as p/q, for positive integers p,q. Let R₁ = p/(2q). R₁ is also a ratio of integers, and hence rational and R₁>0. It is easy to show that R₁ < R. But this contradicts our assertion that R is the smallest rational number.  

hence, there is no smallest rational number greater than 0