Principio definizione ricorsiva

[perplesso1]

Sto leggendo il capitolo introduttivo del testo di topologia di Munkres e mi sono imbatuto in

Principle of recursive definition
Let $ A $ be a set and $ a_0 \in A $. Suppose $ p $ be a function that assigns to each function $ f $ mapping a nonempty section of the positive integers into $ A $ an element of $ A $. Then there exist a unique function $ h:Z_+ \rightarrow A $ such that

$ h(1)=a_0 $
$ h(i)=p(h|{1,...,i-1}) $ for $ i > 1 $


Ora sto provando a fare questo esercizio

Let $ (b_1,b_2,...) $ be an infinite sequence of real numbers. The sum $ \sum_{k=1}^n b_k $ is defined by induction as follows

$ \sum_{k=1}^1 b_k=b_1 $
$ \sum_{k=1}^n b_k = \sum_{k=1}^{n-1} b_k +b_n $

Choose the function $ p $ so that the Principle of recursive definition applies to define this sum rigorously.

Ecco io non ho capito come devo definire $ p $ e $ h $ tale che

$ h(1)=b_1 $
$ h(n) = p(h|{1,...,n-1}) $

[Principe2]

IMHO it's not clear what $f$ is. In particular, what's the meaning of "a non-empty section of positive integers into $A$?

[perplesso1]

$ f $ is a map $ f: {1,2,...,n} \rightarrow A $ . Let $ X $ be the set of functions $ X={f: {1,2,...,n} \rightarrow A | n \in N} $ then $ p: f \in X \rightarrow a \in A $ How can I find the function $ p $ ?


P.S. Il mio inglese fa pena xD