For proofs by well-ordering principle the general template is to consider the negation of some predicate $P(n)$. Then assume the set of all elements that fulfill $\lnot P(n)$, i.e.
$\qquad N = \{ n \mid \lnot P(n) \}$
has a smallest element according to WOP, say $m = \min N$, and if we manage to prove that there is another element $m' \in N$ that is less than $m$ that also negates then we contradict our assumption that $m$ is the smallest element.
My question is that should we be proving some sort of a base case as well for the above mentioned template. As in proving that for some base case, $P(\mathrm{base})$ is true?