Largest prime number

I have been thinking for a long time, yet I cannot discover wether there is a prime number that is greater than all the others. What do you think?

Suppose the number of primes is finite: p₁, p₂, ..., p_k, and let n be the product of all these primes. n - 1 > p_k so is not prime, so there is some i such that p_i divides n - 1. Since p_i also divides n, p_i divides n - (n - 1) = 1, which is absurd. Thus the number of primes is infinite.
http://www.hermetic.ch/pns/proof.htm