I'm trying to prove that If $\left(X,d\right)$ is a perfect polish space and $\epsilon>0$, then there exists $A_1, A_2,\dots$ perfect and nonempty $G_\delta$-sets, such that $A_i\cap A_j=\emptyset$ if $i\not=j$, $\operatorname{diam}\left(A_i\right)<\varepsilon$ and $\bigcup_{i\in \omega} A_i=X$.
I've tried taking a numerable dense subset of $X$ and playing with the balls of radius 1 with center the points of the dense.