Let $R$ be a Noetherian ring, $I\subset R$ an ideal, and $S\subset R$ a multiplicatively closed set. Let $T$ be the associated graded ring $\text{gr}_I(R):=R/I\oplus I/I^2\oplus I^2/I^3\oplus ...$
Since $R$ is Noetherian, $T$ is also Noetherian, there exists $s\in S$ such that $\text{Ann}_T(s)=\text{Ann}_T(as)$ for all $a\in S$.(Indeed, one can choose an element $s$ for which $\text{Ann}_T(s)$ is maximal.) Let $u\in I^r\backslash I^{r+1}$ such that $uas\in I^{r+1}$. How can I conclude $us\in I^{r+1}$? In particular, if $as\in I$ then one should have $s\in I$.
Since $u$ is not in the annihilator, how to use the condition about $\text{Ann}_T(s)$?
Consider the element $u+I^{r+1}\in T$. Since $uas\in I^{r+1}$, we have $(as)(u+I^{r+1})=uas+I^{r+1}=0$, so $u+I^{r+1}\in \text{Ann}_T(as)=\text{Ann}_T(s)$. Thus, $s(u+I^{r+1})=0$, that is, $us\in I^{r+1}$.