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.

That's not true. Euclid's proof does not conclude that this expression is a new prime, it concludes that there exists primes that were not in our original list, that lie between the $k'th$ prime and $p_{1}p_{2}...p_{k} + 1$ (including $p_{1}p_{2}...p_{k} + 1$). Among those extra primes that must exist, $p_{1}p_{2}...p_{k} + 1$ may be one of them, or it may not be. It doesn't matter, either way, there are more primes than listed in our initial list.
We have a counter example as soon as $2*3*5*7*11*13 + 1$.
