In this paper Erdős shows a shorter proof for one of his old results stating that $ s(n) = \prod_{p < n} p < 4^n$ where the product is taken over all primes less than $n$. He also remarks that using the prime number theorem one can show $ s(n)^{\frac1n} \stackrel{n\to\infty}{\longrightarrow} e.$
Can someone here prove this result? It does not seem straightforward to me.
One (crude) attempt I tried was to consider the product $\prod_{i=2}^n \frac{i}{\log{i}} = n!\prod_{i=2}^n \frac{1}{\log{i}}$ which I do not know how to estimate, not to mention that I would then have to argue that it is an asymptotic estimate for $s(n).$
Is there a simple way to show the result about $s(n)$ using the prime number theorem?