Twin prime conjecture proof error

Let $n = 8$. Then all primes less than $8$ are $7, 5, 3, 2$. The product of these is $x = 210$.

$x + 1 = 211$ which is prime, $x - 1 = 209 = 11\times19.$

Your proof most likely stems off of a proof that there are an infinite number of primes. It says that if $\mathbb{P}$ is the set of all primes, and $|\mathbb{P}|<\aleph_0$, then intuitively $$\left(\pm1+p_{|\mathbb{P}|}\#\right)\notin\mathbb{P} \:\text{and}\:\forall p\in\mathbb{P},p\nmid\left(\pm1+p_{|\mathbb{P}|}\#\right)$$ (where $p_n\#$ is the $n$th primorial) implying that $\left(\pm1+p_{|\mathbb{P}|}\#\right)$ is prime, proving that there are an infinite number of primes by reductio ad absurdum. However, since $|\mathbb{P}|=\aleph_0$, $\pm1+p_n\#$ is prime is not necessarily true for arbitrary $n$.

The type of primes that are in the form of $\pm1+p_n\#$ are called primorial primes and you can read more about them here.