For the standard generator $1$ of the additive group $\mathbb Z$, take the intervals
$$
F_n=\{-n,-n+1,\ldots,n\}.
$$
Only the two endpoints change under translation by $1$, and hence
$$
\frac{|(F_n+1)\mathbin\triangle F_n|}{|F_n|}
=\frac{2}{2n+1}\longrightarrow0.
$$
The same follows for every fixed integer translation by iteration. Therefore these sets are a <Følner sequence for the integers>, and \b[$\mathbb Z$ satisfies the Følner condition].

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 4 / c / Solution

{c}

The intervals $F_n=\{-n,\ldots,n\}$ form a <Følner sequence> for the additive group $\mathbb Z$, because
$$
\frac{|(F_n+1)\mathbin\triangle F_n|}{|F_n|}=\frac{2}{2n+1}\longrightarrow0.
$$

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