I'm attempting a proof by induction and, for the inductive step, it would be very useful for me to have some control on a Folner sequence. Indeed, let $G$ be a finitely generated amenable group, fix a finite symmetric set of generators for $G$ and denote by $B_n$ the ball of radius $n$ in the Cayley graph of $G$ with respect of the fixed generators.

Is it possible to find a Folner sequence $(F_n:n\in\mathbb N)$ for $G$ satisfying the following properties?

(1) $F_1\subseteq F_2\subseteq F_3\subseteq \dots$ and $\bigcup_{n\in\mathbb N}F_n=G$;

(2) for all $n\in \mathbb N$ there exists $k$ such that $B_k\subseteq F_n\subseteq B_{k+1}$.