# How to show that a strategy is a SPNE in repeated games

Consider the down below

which I have trouble with solving.

For part 1) I have said that a possible outcome path is to play $$(D,D)$$ in the first round and for all rounds following until $$i \leq 298$$. Afterwards play $$(C,C)$$ forever. If anyone deviates then play $$(D,D)$$ forever. This is since we have that

$$0.01 ( \sum_{i=1}^n 0.99^{i-1} + 9 \sum_{i=n+1}^{\infty} 0.99^{i-1} ) = 1.4 \Leftrightarrow n \leq 298$$

Is this allright? However, then to show that it is a NE in 2) I have trouble with. I thought I could use a Proposition in my book saying that because $$(D,D)$$ is ne in stage game, then the strategy must be a SPNE. Is this alright?

For 3) I know what do just following the same as in 1) but I have no idea how to prove if it is a SPNE or not. Can you help me with that? The strategy is to play $$(D,C)$$ for $$i \leq 92$$ and $$(C,C)$$ aftwards. If anyone devaites, then play $$(D,D)$$ forever.

TIA.