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 $B\_k\subseteq F\_n\subseteq B\_{k+1}$.



Existence of nice Folner sequences
This is not possible for $\mathbb{Z}\wr \mathbb{Z}$ and (most) other groups. For the structure of Folner sets there, see, in particular, Erschler, Anna 
On isoperimetric profiles of finitely generated groups. (English summary) 
Geom. Dedicata 100 (2003), 157–171. and references there. 


