Basic Question about Uniqueness of the Real Numbers

Question

I'm trying to write a proof that $\mathbb{R}$ is isomorphic to any ordered field which has the least upper bound property. I'm glancing at Spivak's Calculus, Chapter 30 when I have issues.

I am trying to prove that $f(m\cdot{}n)=f(m)\cdot{}f(n)$ for the case in which $m,n\in{}\mathbb{N}$. Here is what I tried:

$m,n\in{}\mathbb{N}\implies{}f(m\cdot{}n)=\underbrace{1_F+\ldots{}+1_F}_{m\cdot{}n}=\underbrace{\underbrace{1_F+\ldots{}+1_F}_{m}+\ldots{}+\underbrace{1_F+\ldots{}+1_F}_{m}}_{n}=\underbrace{f(m)+\ldots{}+f(m)}_{n}$

I can't see why $\underbrace{f(m)+\ldots{}+f(m)}_{n}=f(m)\cdot{}f(n)$ to complete the proof. Can anyone help me?

Answer 1

The step you're missing is $$\underbrace{f(m)+\ldots{}+f(m)}_{n} = f(m) \cdot (\underbrace{1_F + \ldots + 1_F}_{n}) =f(m)\cdot{}f(n).$$

Answer 2

$$\underbrace{f(m)+\ldots{}+f(m)}_{n} = f(m)(\underbrace{\bf{1} + \cdots + \cdot 1}_{n}) = f(m)(\underbrace{f(1) + \cdots + f(1)}_{n}) = f(m)(f(\underbrace{1 + \cdots + 1}_{n})) = f(m)f(n)$$