Vtyjkyuk

Let $Gamma$ be a normal form game. An equilibrium $s$ of $Gamma$ is a perfect equilibrium of $Gamma$ if $s$ is a limit point of a sequence $s ( eta ) _eta downarrow 0$ with $s(eta)in E(Gamma,eta)$ for all $eta $ i.e. $s$ is perfect if there exist sequences $s(t)_t inmathbbN$ and $eta(t)_tin mathbbN$ with
$s(t)in E(Gamma,eta(t))$ for all $tinmathbbN
$, and such that $s(t)$ converges to $s$ and $eta(t)$ converges to zero, as $t$ tends to infinity.

My question is: how do I get and precisely understand the switch from $$s(eta) text to s(t)$$ ?

If I understand your question correctly, this is just notational density. You use $eta$ as the perturbation (minimal tremble) of the stratgies. That is $E(Gamma, eta)$ is the set of equilibria of the game $Gamma$ such that all strategies for all agents puts weight at least $eta$ on each action (pure strategy). We call an equilibrium proper if it can be approximated arbitrarily well by such restricted games.

Your sequence $eta(t)$ is just $eta_1, eta_2, ...$, is a sequential relaxation of the perturbation of the game (in the sense of action weights). What is needed then is a corresponding sequence of equilibria $sbig(eta(t)big) in Ebig(Gamma, eta(t)big)$ that similarly converge to $s$. Think of it as approximation: if as a sequence of perturbations subsides to zero, we would wish that there is a sequence of equilibria of the perturbed games that converges (in the space of strategies) to the equilibrium of interest, $s$.

In action, consider the following game.

$$ beginarrayc hline
& L & R \ hline
T & 1,1 & 0,0 \ hline
B & 0,0 & 0,0 \ hline
endarray $$

Clearly $B,R$ is an equilibrium. It is not, however, perfect. This is because $R$ is a best response for the column player if and only if the row player's strategy plays $B$ with probability one. In any tremble, row puts at most probability $1-eta$ on $B$, hence every equilibrium of every perturbed game with $eta > 0$ puts weight $eta$ on $R$ for column, hence as $etato 0$, any sequence of equilibria of perturbed games must converge toward $T,L$, the unique perfect equilibrium of this game.