Subset-sum approximation algorithm running time
Both $z$ and $z'$ belong to the same trimmed list, after trimming. The point is that after trimming, two adjacent elements are at a multiplicative distance of at least $1+\epsilon/(2n)$, by construction. This is why the trimmed lists are small, and so the running time is reasonable. If there is no trimming then there is no problem – the original list already satisfies this condition.

Here is a concrete example. Suppose we want our elements to be at multiplicative distance at least $2$, and suppose that the original list is $1,2,3,4,5,6,7,8$. After trimming we are left with $1,2,4,8$, and any two adjacent elements are at multiplicative distance at least $2$ (in this case, exactly $2$). Had we started with, say, $1,4,9,16$, no trimming would be necessary, but still adjacent elements are at multiplicative distance at least $2$.