I got stuck with this problem while reading Density in Arbitrary Semigroups by Hindman and Strauss. It says:
Problem: If $S$ is a countable semigroup. Then SFC on $S$ implies the existence of a left Følner sequence.
Definition of SFC: $\forall\; H\in \mathscr{P}_f(S),\; \forall\; \epsilon>0,\; \exists\; K\in \mathscr{P}_f(S)$ such that $\forall \; s\in H, \; |K\Delta sK|<\epsilon|K|$.
Definition of left Folner sequence: Let $S$ be a semigroup. A left Følner sequence in $\mathscr{P}_f(S)$ is a sequence $\{F_n\}_{n\in\mathbb{N}}$ in $\mathscr{P}_f(S)$ such that for each $s\in S, lim_{n\rightarrow\infty}\frac{|sF_n\Delta F_n|}{|F_n|}=0$
($\mathscr{P}_f(S)$ stands for the set of all finite subsets of $S$).
Thanks for any help.
Best Answer
Hint Since $S$ is countable, you can construct an increasing sequence $H_1 \subset H_2 \subset ... \subset H_n \subset ...$ of finite subsets such that $$S= \bigcup_n H_n$$
For each $H_n$ you can find some $F_n$ such that $$ \frac{|sF_n\Delta F_n|}{|F_n|} < \frac{1}{n} \qquad \forall s \in H_n $$
Show that $F_n$ is a left Følner sequence.