How can I prove that a set of natural numbers always have a minimum? Let's say I have a finite not-empty set named A, which is a set of natural numbers.
How do I prove it has a minimum?
(In Calculus)

Suppose not. Then, for every $x\in A$ you can find a smaller $y\in A$. Now consider the set of all numbers that are strictly less than every number in $A$. Clearly, $0$ is there (it is $\le y<x$ and if $t$ is there, then $S(t)$ is there ($t<y$ implies $S(t)<S(y)\le x$). Thus, all $\mathbb N$ is there which is absurd. Note that the last step is nothing but the minimalistic version of the "induction" axiom and that I assumed that the ordering is already defined and all relevant fact I used are proven. The axiomatic games are never fun... 
