Show the correctness: $\log^3( n)\in o(n^{0.5})$

Question

show the correctness: $\log^3 (n)\in o(n^{0.5})$?

I started from this way

$$\log \log \log( n) = n^{0.5}$$

then I take $\log$ for two parties

$$\log\log\log\log( n) = 0.5 \log( n)$$

I'm not really know what is the next step.

Could someone help me?

Answer 1

First using L'Hopital or any other method you prefer obtain the result that for any $k>0$ we have: $\lim_{x\to \infty}\frac{\log(x)}{x^k}=0$ so that $\log(x)\in o(x^k)$.

Also observe that if $f \in o(g)$ and $g \in o(h)$ then $f \in o(h)$.

So $$ \begin{aligned} \log( \log(\log(x))) & \in o(\log(\log(x)))\\ \Rightarrow \log(\log(\log(x))) & \in o(\log(x)) \\ \Rightarrow \log(\log(\log(x))) & \in o(x^{1/2}) \end{aligned} $$