So we know that some prime number divides n, we'll name that prime number , and say that the quotient is some integer a (it must be an integer because if it wasn't, then p would not divide n)
right, so (which coincidentally spells pan, but that's not part of the proof)
Now, either or a is < because if neither was, then both would be greater than the And if they were both greater than the we could say that > n However, and so this is a contradiction.
This means that either or a is not greater than therefore either or a must be less than .
If is less than then there is a prime number which divides n and is less than the . If is greater than , then a must be less than . And by the Fundamental Theorem of Arithmetic, a is either prime, or the product of two or more primes, so either a or one of its factors is prime. So there is a prime number less than which divides n.
It's also worth pointing out that there is a theorem which says if a divides b, and b divides c, then a divides c. This is for if your instructor wants you to show that if a is less than the , but a is not prime, then one of a's prime factors will divide n.