I am not being able to find what is wrong in this proof.
statement: For any set of rationals there is a least element in the set.
Hypothesis: $p(k)$=For set of k rationals there exist a least element in the set.
It is trivial to prove that p(1) and p(2) are true.
Now suppose $p(k)$ is true. For Every set of k rationals there is a least element in the set.
Now we check truth value of $p(k+1)$ .
as we can split set $k+1$ of rationals as set of $k$ and 1 rational.
Now , as we know both of them have least element hence least among them will be least element in the set.
Hence,$$p(k)\implies P(k+1)$$ and Hence, Using PMI we prove that any set of rational elements has a least element .
Which is not true for rationals belonging in (0,1).