Prove: $(\log{n})^k=\mathcal{O}(n^{\frac{1}{\mathcal{m}}})$ $k,m\in\mathbb{N}$

Question

($\mathbb{N}$ does not contain 0 in this case)

How can I prove this? It feels intuitive and seems to be right. Sadly, I got no idea how to approach this proof. Can someone give me advice? There is a tip on the worksheet, which may be used:

$\lim_{x\rightarrow\infty}\frac{(\log{x})^k}{x}=0$

Thanks in advance!