Finitely repeated Prisoner’s Dilemma with switching cost
A couple hints.

Regarding the lower bound on $\epsilon$: What happens if deviation occurs at stage $T$? In other words, there is no opportunity for your so-called &quot;punishment stages&quot;.
Regarding the upper bound on $\epsilon$: Suppose player 2 deviates at stage $T-1$ but player 1 does not. What must be true about $\epsilon$ in order for player 1 to have an incentive to switch to $D$ at the very last stage?


In general, to show that the given strategy profile is subgame perfect, you need to argue that playing $C$ after $(C,C)$ and $D$ after either $(C,D),(D,C),(D,D)$ are optimal.