This is a proof I read in a paper ().
One of the key conditions the paper wants to prove is below \begin{equation} s_{n}^{-2}\sum^{n}_{i=2}E\{Y^{2}_{ni}I(|Y_{ni}|>\varepsilon s_{n})\}\rightarrow 0, \end{equation}as $n\rightarrow \infty$ for each $\varepsilon>0$.
It proves a condition \begin{equation} s_{n}^{-4}\sum^{n}_{i=2}E(Y_{ni}^{4})\rightarrow 0, \end{equation} and claims that the latter condition implies the former condition. Can someone explain to me why？ Thanks.
Notice that \begin{align} Y_{ni}^2I(|Y_{ni}|>\varepsilon s_n)&= \varepsilon^{-2}s_n^{-2}\cdot Y_{ni}^2\varepsilon^2s_n^2I(|Y_{ni}|>\varepsilon s_n)\\ &\leqslant \varepsilon^{-2}s_n^{-2}\cdot Y_{ni}^2Y_{ni}^2I(|Y_{ni}|>\varepsilon s_n)\\ &\leqslant \varepsilon^{-2}s_n^{-2}\cdot Y_{ni}^4, \end{align} then take the expectation.