Every composite integer has a divisor less than or equal to its square root. But why?

Proof

Any composite integer N can be written as $ N = a * b $ where $1 < a, b < N$.

At least one of these two divisors a and b must be less than or equal to $\sqrt{N}$.

Let’s assume that both a and b are greater than $\sqrt{N}$. Then $a*b > N$. That is a contradiction. Therefore it must be the case that at least one of the divisors is less than or equal to $\sqrt{N}$.