real analysis – definition of a sequence

I am having trouble understanding this definition. Can someone help me understand this through examples? I’ve tried reading it slowly, several times and I can’t determine what it is actually saying! It doesn’t seem like there is anything inductive or recursive about this definition. Where does the definition or theorem say anything about the value of $x_n$ being dependent on a function of $x_{n-1}$? Is this unique element $x_1$ a member of the $x_{s(n)}$ sequence? What does the $x_{s(n)}$ even mean? I just basically do not understand this defintion and theorem, so please help me.

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