Why do we need a restriction of a game to prove the given statement?

Consider a game $$G$$. We have to prove that is $$s$$ is a Nash Equilibrium of $$G$$, then it is also a Nash Equilibrium of the game formed by removing strictly dominated strategies of $$G$$.

I looked at the proof for this in https://homepages.cwi.nl/~apt/stra/ch3.pdf (the proof is described in first two pages of it) They use a restriction $$R$$ ($$R_{i}=$$ (possibly empty) set of strategies such that $$R_{i} \subseteq S_{i}$$) of a game at every point of the proof.

My question is what good does it do to take the $$R$$ of the game instead of directly taking the set of all possible strategies for all the players $$S$$?

The point of the proof is to show the result for ANY $$R$$ such that $$R \rightarrow_{S}R'$$.
In $$S$$, there is only one strictly dominated strategy, and we remove it and get $$R$$ (hence $$S \rightarrow_S R$$). Now look at $$R$$. There is a strictly dominated strategy that was not one before (in $$S$$), removing it get $$R'$$ (hence $$R \rightarrow_S R')$$.