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## Homework Statement

This theorem comes from the book "The Real Numbers and Real Analysis" by Bloch. I am having a hard time understanding a particular part of the proof given in the book.

__Prove the following theorem:__

There is a unique binary operation +:ℕ×ℕ→ℕ that satisfies the following two properties for all n,m ∈ ℕ.

1) n+1=s(n)

2)n+s(m)=s(n+m)

## Homework Equations

The following is info that can be used to prove the theorem

.

__Definition__: Let S be a set. A binary operation on S is a function S×S→S.

__Axiom__: There exists a set ℕ with an element 1 ∈ ℕ and a function s:ℕ→ℕ that satisfy the following three properties

1) There is no n ∈ ℕ such that s(n) = 1

2) The function s is injective (one-to-one)

3)Let G⊆ℕ be a set. Suppose 1 ∈ G, and that if g ∈ G then s(g) ∈ G. Then G = ℕ

__Lemma:__Let a ∈ ℕ. Suppose a ≠ 1. Then there is a unique b ∈ ℕ such that a = s(b)

__Theorem (Defintion by Recursion):__Let H be a set, let e ∈ H and let k: H→H be a function. Then there is a unique function f: ℕ→H such that f(1)=e, and f(s(n))=k(f(n))

## The Attempt at a Solution

The following is the proof given in the book for existence of such an operation . The underlined bolded part is what I am having trouble understanding.

Suppose p ∈ ℕ. We can apply the theorem above to the set ℕ,

__and the function s: ℕ→ℕ to deduce that there is a unique function f__

**the element s(p) ∈ ℕ**_{p}:ℕ→ℕ such that f

_{p}(1)=s(p) and f

_{p}(s(n)) = s(f

_{p}(n))...

Why is s(p) ∈ ℕ? I do realize the function s is defined as s:ℕ→ℕ, so the output of the function s (if it exists) must be in ℕ, but how do we know s(p) is defined from the above info or from the definition, axiom, or theorem above. The only reason I can think of is we can assume for any p that s(p) is defined, but I am not conviced because that axiom only says the function is injective, not bijective,

Any help would be appreciated,

Thank you,

Ethan