real analysis - Proof $frac{log(n!)}{nlog n}$ is increasing for positive integer $n$ ($log n=ln(n)$)

I would like to show that $\frac{\log(n!)}{n\log n}$ increases as n increases, for positive integer n only (to ignore the use of gamma function). Note here $\log n = \ln(n)$, so using base e. I would like to be able to show this so I can then apply the Monotone Convergence Theorem to show that $(\log(n!)\sim(n\log n)$. My first idea was to try and show that the second derivative is always positive, but due to $\log(n!)$ not being continuous (without having to use gamma function which I feel is OTT for this question) this, of course, would not work. Does anyone have any ideas about how to prove that this sequence is increasing rigorously?