I answered a question to prove that there are infinitely many prime numbers, but I'm not sure if my attempt is right. Can somebody help me to check if my attempt is right? I would like, if I am wrong, not a answer, but just a hint, thank you to whoever helps me!
Question: Prove that the sequence of prime numbers is infinite using the sequence $\{R_n\}_{n\geq1}$ given by $R_n = n! + 1$.
My attempt:
Suppose that the sequence of prime numbers is finite, so the sequence $R_n$ is finite, therefore, $R_{n+1}$ is not a prime number, in other words, $R_{n+1} = t*k$, $t,k \in \mathbb{Z}^+$, but $R_{n+1} = (n+1)! + 1$ by definition of the sequence $R_n$.
Note that $R_{n+1} = (n+1)n(n-1)! + 1$ and $6|n(n+1)$, so $R_{n+1} = 6v(n-1)! + 1, v \in \mathbb{Z}^+$
The only integer number that divide 1 is himself, therefore, $1|R_{n+1}$, because $1|6v(n-1)!$ and $1|1 \Longrightarrow 1|6v(n-1)! + 1$.
Furthermore, $R_{n+1}|R_{n+1}$, therefore, $R_{n+1}$ is a prime number, this is a absurd, because $R_{n+1}$ is not a prime number by hypothesis, therefore, $R_{n+1}$ is a prime number and then the sequence of prime numbers is infinite.