Proving the Recursion Theorem (Hungerford) The proof below was taken from Hungerford.  I can see how $R = \cap_{Y \in G} \, Y$ defines the desired $\varphi:  \mathbb N \rightarrow S$, but I don't see why he had to go through so many mechanics to do so.
I can define $\varphi:  \mathbb N \rightarrow S$ as follows:
\begin{align}
\varphi(0) & = a \\
\varphi(1) & = f_0(\alpha) = f_0(\varphi(0)) \\
& \vdots \\
\varphi(n) & = (f_{n-1} \circ f_{n-2} \ldots \circ f_0)(\alpha) = f_{n-1}(\varphi(n-2))
\end{align}
Then at $n + 1$, we have:
$$\varphi(n + 1) = (f_n \circ f_{n-1} \ldots \circ f_0)(\alpha) = f_n((f_{n-1} \circ f_{n-2} \ldots \circ f_0)(\alpha)) = f_n(\varphi(n))$$
So, inductively I did the same thing.  In fact, I got the clue from the highlighted text.  Was the author just proposing a different proof, or what is being gained in his alternative proof? 



 A: This is a common phenomenon. Hungerford is giving a "top-down" construction of the function, while you are giving a "bottom-up" construction. Both methods work. Note that Hungerford's method does not require writing a formula for an arbitrary value of $f$, while your method does. 
One reason that Hungerford might choose this method is that the "top-down" approach is used in several other circumstances in algebra:


*

*Defining the subgroup of a group $G$ generated by some set $H \subseteq G$.

*Defining the subfield of a field $F$ generated by some set $H \subseteq F$.


and similarly for generated subsets of rings, vector spaces, etc.  These can be defined in a "bottom up" way, by stating the explicit form of an element of the generated object, but it is often more elegant to use a "top down" definition. For example, the subgroup of $G$ generated by $H$ is defined to be the intersection of all subgroups of $G$ that contain $H$. 
In other areas of mathematics, there is the same option of defining things "from the top" or "from the bottom". A key example is the class of Borel sets on a topological space. The "bottom up" characterization of Borel sets is more complicated, and so the top-down characterization is used more often in mathematics textbooks. 
