Let $p_n$ be the $n$th prime in ascending order. We wish to prove that $p_{n+1} - p_n < p_1 ... p_n$
Proof: Observe $N = p_1 ... p_n + 1$ which by the division algorithm is not divisible by any $p_i$ for $i = 1, ... , n$. We know $N$ must have some prime divisor. Let $p_k$ be the prime divisor of $N$ where $p_k > p_n$. Thus, $p_{n+1} \leq p_k \leq N = p_1 ... p_n + 1$ Since $p_n > 1$, $p_{n+1} - p_n < p_1 ... p_n$ Therefore, the difference between the $n+1$st prime and the $n$th prime is less than $p_1 ... p_n$, and so there are infinite primes.