The Prime Numbers Set is infinite. Is this proof correct?

Question

Proposition: The Prime Numbers Set is infinite.

Proof: Suppose we have a finite set of prime numbers $p_{1}, p_{2}, ..., p_{n}$ such that $p_{n}$ is the largest of them.

Define $ c := p_{1}*p_{2}*...*p_{n}$

$c$ is clearly not prime.

Let $q = c + 1$ such that q is not divisible by $ p_{k} $ or other element of the set since 1 is not divisible by them.

Then $q$ is by itself a prime number or it is divisible by a prime number greater than $p_{n}$. That is not possible, since $p_{n}$ is the largest prime number. Contradiction!

Q.E.D.

Answer 1

Hint: http://en.wikipedia.org/wiki/Euclid's_theorem#Euclid.27s_proof

Answer 2

The proof is technically correct, but it lacks clarity, much like many other proofs I have read.

Suppose the list of primes is finite and denoted as $\{p_1, p_2, ..., p_n\}$. Now define:

$q = p_1 \times p_2 \times ... p_n + 1$.

If $q$ is a prime number, then it is certainly not in the list, which contradicts our assumption that the list of primes is finite.

If $q$ is not a prime number (thus a composite number), it must have prime factors. However, these prime factors cannot be any of the primes in the list, because dividing $q$ by any of them would always yield a remainder of $1$, as shown below:

$q = (p_1 \times p_2 \times ... p_{i-1} \times p_{i+1} \times ... p_n) \times p_i + 1$

This, again, contradicts our assumption. Q.E.D.