So I am reading a paper and I came across the following statement
\begin{equation} \inf_{\epsilon > 0} \bigg\{4\epsilon + \frac{24}{\sqrt{n}(2-p)\epsilon^{\frac{p}{2}-1}} \bigg\} = \mathcal{O}\bigg(\frac{1}{n^{\frac{1}{p}}}\bigg), \end{equation}
as $n\rightarrow \infty$ where $p>2$ and $n \in \mathbb{N}$. So my question is what is the general idea behind proving statements like this, involving $\inf$ and big-O. I tried to find the $\inf$ of this function on the left side through differentiation, so that I get rid of the dependence from $\epsilon$ but I don't think it's easy to do that. Any ideas on how to approach this? Thanks!
By substituting $\epsilon=1/n^{1/p}$ we see that \begin{equation} \inf_{\epsilon > 0} \bigg\{4\epsilon + \frac{24}{\sqrt{n}(2-p)\epsilon^{\frac{p}{2}-1}} \bigg\} \leq \left(4+{24\over 2-p}\right) {1\over n^{1/p}}= \mathcal{O}\bigg(\frac{1}{n^{\frac{1}{p}}}\bigg), \end{equation}