Let $G(n)$ denote the largest prime gap $p_{k+1}-p_k$ occuring between $1$ and $n$.
By considering the $n-1$ composite consecutive integers $n!+2$,...,$n!+n$ we can conclude that $$G(n!+n) \geq n-1 $$
Then, as mentioned on this post, it follows (without any proof) that by using Striling's approximation we get $$G(n) \gg \frac{\log(n)}{\log(\log(n))}$$
I tried at least to pass from $G(n!+n)$ to $G(n!)$ by considering the sequence $(n-1)!+2$,...,$(n-1)!+n-1$ and noting that the last term is smaller than $n!$ thus obtaining $$G(n!) \geq n-2$$
Now I am stumped as to how I should use Stirling's approximation. I believe I need to use the inequality $ log(n!) \leq nlog(n)$ as I read on another website (but also without any further explanation).
I would like to have a $G(n)$ instead of $G(n!)$ but I also don't know how.
How should I proceed from here?