I want to solve this proof by the method of Contradiction. Though without using the well ordering principle. I don't have any idea how to start. I have found other ways to prove this theorem but only by using the well ordering principle. So is it even possible to solve without using the well ordering principle?
P.S. I'm new so any advice on formatting or setting up a question would be much appreciated. Thank you.
Suppose for a contradiction that $$m<a\quad\text{ but }\quad m\not\leq a-1$$ (Assume that $a\geq2$ since otherwise $a-1$ is not a natural number.)
Note that $m\not\leq a-1$ is completely equivalent to $m>a-1$. Combining those inequalities we get the following: $$a-1<m<a$$ This is a contradiction because you cannot fit a natural number strictly between two adjacent natural numbers. In particular since $m<a$ we have that $a-m$ is a natural number, and from $a-1<m$ we can obtain $$a-m<1$$ which contradicts the well-ordering principle.