Prove or disprove that $log^{k}(n) \in O(\sqrt{n}) \forall k > 0$

Question

I'm trying to solve the problem described in the title. By using the free version of wolfram and testing some increasing values of $k$ I get that:

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

And apparently $log^{k}(n) \in O(\sqrt{n})$, but by trying to solve this limit on paper in order to reach an appropriate proof I would need to keep applying L'Hospital rule indefinitely. Is that what I suppose to be doing? How could I proceed to build this proof?