Feller condition and Chebyshev inequality

Question

So according to https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1756450. Feller condition of the CIR process can be proven by such a way (it is quite short (2 pages)). But towards the ending of the proof, I am sort of got lost. Suppose that $r(t)$ is a CIR process. From page 2 of the paper (at the Gronwall inequality), we get it inequality as \begin{equation} E(r^{-m} (\tau_\varepsilon \land t)) \leq r^{-m}(0) exp(mkt) \end{equation} What the point of the above inequality? Then how do I apply the Chebyshev inequality to get the following inequality (last equation of page 2) \begin{equation} P(r^{-m}(\tau_\varepsilon) \geq \varepsilon^{-m}) \leq \varepsilon^m r^{-m}(0) exp(mt) \end{equation} Thanks in advance!