How do we know Euclid's sequence always generates a new prime? How do we know that $(p_1 \cdot \ldots \cdot p_k)+1$ is always prime, for $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $\ldots$ (i.e., the first $k$ prime numbers)?
Euclid's proof that there is no maximum prime number seems to assume this is true.

As others have mentioned, the proof you are referring to is probably the proof by contradiction that there are infinitely many primes. Compare with the following statement:

If there are finitely many non-square numbers, then there is a largest non-square number $n$ and so $n+2$ is a square.

One can see that the above statement is true, but that certainly does not imply that if we start with any non-square number $x$, we would get $x+2$ to be a square number!
