A Fantabulous integer is an integer which has another fantabulous integer smaller than it BdMO 2013 problem-7:

A positive integer is called “Fantabulous” if there is another fantabulous positive integer smaller than it. Find the 
  number of fantabulous integers.

I am bamboozled at this question.Nevertheless:1 cannot be a fantabulous integer as it has no positive integer smaller than it.But 2 isn't a fantabulous integer either,as it has only one integer smaller than itself,and that is 1.But we know that 1 is not a fantabulous integer.By similar reasoning,we can prove that no fantabulous integer exists.We prove the preceding statement through contradiction.
PROOF: Let $F$ be the set of fantabulous integers.Assume F is non-empty.By the well-ordering principle,there exists a smallest integer $f$ in the set.But by definition,there exists a smaller fantabulous integer in the same set,contradicting our claim that $f$ is the smallest fantabulous integer.Thus no fantabulous integer exists.
Am I right?Also,if anyone finds any suitable tag,feel free to edit.
 A: 
A positive integer is called “Fantabulous” if there is another fantabulous positive integer smaller than it. Find the 
  number of fantabulous integers.

Below is the required proof of this theorem,copied from my question[Credits to Prahlad Vaidyanathan for verification of this proof and Hagen Von Eitzen for a correction]
PROOF: Let $F$ be the set of fantabulous integers.Assume F is non-empty.By the well-ordering principle,there exists a smallest integer $f$ in the set.But by definition,there exists a smaller fantabulous integer in the same set,contradicting our claim that $f$ is the smallest fantabulous integer.Thus no fantabulous integer exists,which implies that $F$ is empty.
Q.E.D
The fact that no fantabulous integer exists can also be proven via induction[credits to N.S. and brusko651]
BASE CASE: 1 is not fantabulous because no positive integer smaller than 1 exists.2 is not a fantabulous integer either,since the only positive integer smaller than it is 1,which is non-fantabulous.
INDUCTIVE STEP: Suppose $n$ is not a fantabulous integer.We prove that $n+1$ is not a fantabulous integer as well.
If $n+1$ is fantabulous,then by definition,there must exist another integer smaller than $n+1$ which is fantabulous.The greatest positive integer smaller than $n+1$ is n.But we know that $n$ is not a fantabulous integer,and from that,it follows that no fantabulous integer smaller than $n$ and thereby $n+1$ can exist.Thus $n+1$ is not a fantabulous integer.
Q.E.D
