Some preliminary definitions. Let $G$ be a graph, and let $X,Y\subseteq V(G)$ be sets of vertices in $G$. Let $$e(X,Y):=\bigl|\{(x,y)\in X\times Y:xy\in E(G)\}\bigr|,$$ and define the edge density between $X$ and $Y$ by $$d(X,Y):=\frac{e(X,Y)}{|X||Y|}.$$
We call the pair $(X,Y)$ $\epsilon$-regular if, for every $A\subseteq X$ and $B\subseteq Y$ such that $|A|\ge\epsilon|X|$ and $|B|\ge\epsilon|Y|$, we have $$\bigl|d(A,B)-d(X,Y)\bigr|\le\epsilon.$$
Notice that $\epsilon$-regularity holds trivially with this definition for $\epsilon\ge1$, so we only have to consider $\epsilon<1$.
Question. How do I prove that regularity is inherited? More precisely, if $(X,Y)$ is an $\epsilon\eta$-regular pair, then I want to show that $(X',Y')$ is $\epsilon$-regular for all $X'\subseteq X$ and $Y'\subseteq Y$ with $|X'|\ge\eta|X|$ and $|Y'|\ge\eta|Y|$.
What I’ve tried. It is easy to prove this when $\eta<1/2$, since we can use $\epsilon\eta$-regularity to show that $$\bigl|d(X',Y')-d(X,Y)\bigr|\le\epsilon\eta \quad\text{and}\quad \bigl|d(X'',Y'')-d(X,Y)\bigr|\le\epsilon\eta;$$ here $X''\subseteq X'$ and $Y''\subseteq Y'$ with $|X''|\ge\epsilon|X'|$ and $|Y''|\ge\epsilon|Y'|$. Also, when $\eta\approx1$, then $X'$ and $Y'$ constitute large portions of $X$ and $Y$, so the densities of $(X,Y)$ and $(X',Y')$ should be similar. We can compute $$d(X,Y) =\frac{e(X,Y)}{|X||Y|} \le\frac{e(X',Y')+(1-\eta)^2|X||Y|}{|X||Y|} \le d(X',Y')+(1-\eta)^2$$ and $$d(X',Y') =\frac{e(X',Y')}{|X'||Y'|} \le\frac{e(X,Y)}{\eta^2|X||Y|} =d(X,Y)+\Bigl(\frac{1}{\eta^2}-1\Bigr)d(X,Y),$$ but I'm not sure what to do from here.
How should I approach this problem when $1/2<\eta<1$? Thank you.