Theorem: If n is a composite natural number then n has a prime divisor less than or equal to the square root of n.
Proof: Suppose that n = ab for nonzero numbers a and b. If a is composite then its prime factors are clearly less than a; similarly for b. Let's suppose that both a > sqrt(n) and b > sqrt(n) and see if this leads to a contradiction:
which contradicts the fact that n = ab. This means that either a or b is less than the square root of n, which in turn means that n has a prime factor less than its square root. This finishes the proof.