You are overcomplicating the question! Assume the primes less than the squareroot of n do not divide n, and n is not prime. Then there is some prime p dividing n. By assumption, p must be larger than the square root of n. But n=pq for some integer q, and clearly q is less than the square root of n. Either q is divisible by a prime less than the square root of n, or q=1. In the first case we contradict our assumption since x divides q implies x divides n. In the second case, p=n is prime, which again contradicts our assumption.