In between Primes

Since we know $n$ is the product of all the prime numbers from $1$ to $p$ , so we can say that the prime factorization of $n$ is $1\cdot2\cdot3\cdot5...p$ . If $n-1=m$ we can say that $m$ is relatively prime to all prime numbers from $1$ to $p$ , and if $m$ is relatively prime to all potential prime factorization, then $m$ must be prime itself, and we know $n-1=m$ satisfies the condition $p<m<n$ , thus showing it is always possible .