Let $A$ be a nonempty, finite, simply ordered set.
We have just proved that every such set has a largest element and we are now out to prove that such a set is of the same order type as some section of the positive integers (this is problem 4 of section 6 of Munkres for those who'd like a reference).
The most natural way I thought to prove it was as follows:
Since $A$ is a nonempty, finite, simply ordered set, we may write $A$ as $a_1<'a_2...<'a_k$ where $<'$ denotes the order relation on $A$.
We then define a function $f:A\rightarrow$$\Bbb N_k$ where $\Bbb N_k$ denotes the first k positive integers by $f(a_i)=i$ for each $a_i$ in $A$
To insure that $f$ is a function I supposed $a_x$, $a_y$ $\in A$ with $x=y$ and said that by our definition of $f$ we have that $f(a_x)=x$ and $f(a_y)=y$ and since $x=y$ we have $f(a_x)=f(a_y)$ and therefore $f$ is a function.
Suppose $a_a, a_b\in A$ with $a_a<'a_b$ where $<$ denotes the order relation on A. Then $a, b\in \Bbb N$ with $a<b$ as seen above by the given ordering of $A$ seen above where $<$ denotes the usual ordering on $\Bbb N$. By the definition of f $f(a_a)=a$ and $f(a_b)=b$ and $a<b$ as previously noted and so $f$ preserves the order relation.
To show injectivity, suppose $f(a_x)=f(a_y)$. Then $x=y$, hence $a_x=a_y$ , and we have that f is injective.
Lastly, to show that f is surjective, let $m\in\Bbb N_k$. Then $m<k$ by the definition of $\Bbb N_k$, and so $f(a_m)=m$ with $a_m\in A$ and hence $f$ is surjective.
I was wondering if this proof is correct, because after doing it, I checked it against some online solutions none of which proved it in the same manner (they mostly used induction). I would like to know if my way is just another way of proving it or if there is something wrong with it. For example, I am suspicious as to whether my assumption that we could order $A$ in such a way is justified. We have no theorem in the book that says a set with an order relation may be written in such a way. Also, I guess that I assumed that $A$ has a least element. Was that something I should have proved as well? Other than that, I think I was solid, but please don't let if you see anything else.
In summary, I would like to know if this is a proper proof.
Many thanks :)