Is there an infinite number of primes constructed as in Euclid's proof?

In Euclid's proof that there are infinitely many primes, the number $p_1 p_2 ... p_n + 1$ is constructed and proved to be either a prime, or a product of primes greater than $p_n$.

Trivially, we could also use the number $R_n=p_1 p_2 ... p_n - 1$ to prove the theorem, for n>2.

Intuitively, as $n$ grows, the probability that $R_n$ is prime gets smaller. Is there a proof that $R_n$ is not prime for any $n$ greater than some integer $M$ ? Or conversely, that there is an infinite number of prime $R_n$ numbers?

A possibly equivalent question: Is there a prime number greater than $p_n$ and smaller than $R_n$ for any $n$ greater than some integer $M$ ?

The answer to the last question is yes. Much more is known: there is always a prime between $p$ and $2p$, by Bertrand's Postulate, which has long been a theorem.

The numbers $P_n=p_1p_2\cdots p_n +1\:$ have been looked at quite a bit, your $R_n$, for no clear reason, somewhat less. Very little of a general character is known about the $P_n$, and even less about the $R_n$. Prime numbers of either shape are called primorial primes.

It is not known whether there is an infinite number of primorial primes. More startlingly, it is not known whether an infinite number of the $P_n$ or $R_n$ are composite! There has been a great deal of computational work on these numbers, and primes do seem to become scarce among them as $n$ gets large. You can find some information here.

Euclid's proof says that if there were only finitely many primes, the product of these primes +1 would be a new prime. It arrives at a contradiction.

Hence, it really makes no assertion about the product of the first $n$ primes, plus one.