Prove that $\log \log n/ \log n$ goes to zero with almost elementary techniques

Question

I am trying to show that:

$$ \lim_n \frac{\log \log n}{\log n} = 0 $$

without using "advanced" calculus techniques (derivatives, analysis of the function $x \mapsto \log \log x/ \log x$…). I was able to show that the sequence

$$ n \mapsto \log n / n $$

is monotone decreasing, but this does not seem to trivially imply that $n \mapsto \log \log n / \log n$ is also monotone decreasing. Is there any way around this? Once monotonicity is proven, one can argue that $\log \log n / \log n$ is positive and eventually smaller than every (positive) number and/or find a suitable subsequence converging to zero. Thank you in advance.

Answer 1

Let $f(x)=(\log x)/x$ and $g(x)=\log x$. Then $(\log\log x)/\log x = f(g(x))$ is the composition of a decreasing function and an increasing function; and such a composition is automatically decreasing.