 At the end of stage $i$, $[f]$ was restricted to vary in some affine subspace of codimension $\epsilon k$ of $A_{i-0}$. This affine subspace was chosen by restricting the values of $f$ at $i$. Thus $[f(i)]$ was allowed to vary in a random affine subspace of codimension $\epsilon k$ in $V$. Call this subspace $U$. 