Two integers are relatively prime if there greatest common denominator is 1. The definition of prime is that p can only be divided by 1 and itself. So every integer less than p does not divide p hence their gcd = 1 and hence they are relatively prime.

If you want an proper proof... Might look like this...

Let p be prime, then if p = ab then either a or b = 1. If a=1, b=p and if b=1, a=p.

Let be an integer less than p that divides p, hence p = kl for some integer l. Since ,l must be equal to 1. Hence k must be equal to p which contradicts our statement the k is an integer LESS THAN p that divides p, hence there doesnt exist an integer less than p that divides p.

Im not that good at proofs but i think that is the general idea...