OK, to complete your proof you have to show that if a number doesn't have a prime divisor less or equal than it's prime
Indeed, suppose it doesn't have have a prime divisor less or equal than , then n has at least two prime divisors (if it's composite) such that .
But so the product and this is absurd
So every composite number has at least one prime divisor such that
But you can only achieve the equality if is a perfect square, and in that case so the inequality holds
Otherwise it's not a perfect square and it has a prime divisor such that
So is a divisor of and therefore