I am trying to solve exercise two from the following document : http://mtm.ufsc.br/~daemi/soficworkshop/Course%20notes/Lupini%20Lecture%202.pdf
I suspect there is an error in the exercise, but I'm not sure.
I tried to solve the reverse implication as follows:
Choose $\varepsilon>0$ and a finite subset $F\subset \Gamma\setminus 1$. By assumption there exists an $\eta> 0$, a function $r:\Gamma\rightarrow (0,1)$, an $n\in\mathbb{N}$ and a map $\phi:\Gamma\rightarrow S_n$ such that for all $g,h\in \Gamma$
1) $d_{S_n}(\phi(g)\phi(h),\phi(gh))<\eta$,
2) $l_{S_n}(\phi(g))>r(g)$.
Now define $\psi:\Gamma\rightarrow S_{n^k}: g\mapsto \phi(g)^{\otimes k}$ ($k$ will be fixed later). Clearly $\psi(1)=1$. Using the fact that $l_{S_n}(\phi(g))> r(g)>0$ for all $g\in F$ and the fact that $\min_{g\in F}l_{S_n}(\phi(g))>0$, one concludes (using the hint) that $l_{S_{n^k}}(\psi(g))>1-\varepsilon$ for $k$ large enough. Hence take such a $k$.
It remains to show that
$d_{S_{n^k}}(\psi(g)\psi(h),\psi(gh))<\varepsilon$ for all $g,h\in F$.
However, using the inequaltities I'm only able to find that
$d_{S_{n^k}}(\psi(g)\psi(h),\psi(gh))<1-(1-\eta)^k$.
Since I only have control over $k$ and not over $\eta$, I can not get this as small as I want.
Also, for the forward implication, it bothers me that the function $r$ exists independently of the chosen finite set $F$.
Help would be much appreciated.
Thanks.
Derek Holt answered this question in his comment.