Courant & Robbins' style in that book was to reserve the letter $n$ in an inductive argument for the general case, like this:
We want to prove that some statement $P(n)$ is true for all natural numbers $n$.
First we prove that it is true for $P(0)$.
Now we assume it is true for some natural number $r$, and we will prove it must also be true for $r+1$. That is, we will show $P(r)\implies P(r+1)$.
Here the authors are trying to make explicit the idea that $n$ represents any natural number, and in their induction step they are choosing one particular natural number $r$. This is simply the way that induction is introduced in the book at the top of page 11.
The essential idea in the preceding arguments is to establish a general theorem $A$ for all values of $n$ by successively proving a sequence of special cases, $A_1, A_2, \dots$ . The possibility of doing this depends on two things: a) There is a general method for showing that if any statement $A_r$ is true then the next statement, $A_{r+1}$, will also be true. b) The first statement $A_1$ is known to be true.
Once you are comfortable with this idea, many people simply use $n$ in the second step as well, remembering that in that second step $n$ represents a particular natural number.