I was wondering if I could get some insight on my proof. I am in the midst of relearning some number theory and just "writing proofs" in general, and I would like some assistance to see if I am on the right track.
The statement I am proving is Euclid's Theorem which states that "there are infinitely many primes."
Here is my attempt at the proof after some reading (keep in mind, I am still somewhat of an amateur when using LaTeX so please bear with me!):
Proof. Suppose in order to derive a contradiction there are finitely many primes. That is, we have a complete list $p_1, \dots, p_n$. Let $p$ be the product of all the primes in this list i.e. $p = p_1 \cdots p_n$. Consider the number $$N = p + 1.$$ Since $N > p_i$ for all $i$, $1 \leq i \leq n$, there is no way $N$ can be any of the $p_i$. So $N$ must be composite. By the Fundamental Theorem of Arithmetic, $N$ is a product of primes. So there is a prime, say q, that divides $N$, and $p$ as well. So it follows this $q$ must also divide $$N - p = 1,$$ but this is impossible. No number, or prime, divides 1. Thus, contradicts the assumption that our list is complete and so there must be infinitely many primes.
Any feedback would be appreciated. I always had trouble understanding this theorem and always forgot the "key argument". Now I feel like I finally get it... Hopefully. Thank you for reading!