How do you show $ \lim_{n\to \infty} {\frac{\ln(1)+\ln(2)+\ldots+\ln(n)}{n}} = \infty$? $$ \lim_{n\to \infty} {\frac{\ln(1)+\ln(2)+\ldots+\ln(n)}{n}} = \infty$$
I know I should show that it is greater then something that approaches to $\infty$ but I don't see what.

Suppose $n$ is even. Then in the numerator, at lease $n/2$ of the terms are $>\ln (n/2).$ Thus
$$\frac{\ln 1 + \cdots + \ln n}{n} > \frac{(n/2)\ln (n/2)}{n} = (1/2)\ln (n/2) \to \infty$$
as $n\to \infty$ through even $n.$ This gives the idea; handling odd $n$ is a small step from the above.
