The movie titled ''The Madness of King George III'' was not a sequel to an earlier film called ''The Madness of King George II'', but we all know that not only kings, but movies too, sometimes use postnominal numerals in that way. And similarly with the subject line of this present question, since an earlier one that I posted is as follows:
Illustrative examples of a phenomenon in the logic of mathematical induction
Mathematical induction as taught to undergraduates almost always says one starts with a "base case."
There is a kind of mathematical induction that includes no base case, and that is this:
\begin{align} & \text{Prove that if for every natural number $m$ less than $n,\, P(m).$} \tag 1 \\ & \text{Conclude that for every natural number $n,\, P(n).$} \tag 2 \end{align}
No base case is needed because it is vacuously true that for every natural number $m$ less than the smallest natural number, $P(m),$ and so the base case is proved by line $(1)$ above.
So here is my question: Is there some variety of examples (or, at least, are there some examples) of problems that call for this form of induction and that are suitable for undergraduates being taught to understand what induction is and how it can be used? ("Suitable" might be taken as excluding theory of ordinal numbers in set theory since that would require an unduly large amount of exposition to those unfamiliar with it when a course must cover a fair number of other things.)
This kind of induction is often used in elementary number theory. Since in general there's no direct relation between multiplicative structure of $n-1$ and $n$, the "traditional" induction is often useless.
Instead, one may use induction about the smaller factors of a number to prove a property of a number.
Here's a proof that every natural number $n\ge 2$ is a product of primes.
If $n$ is a prime, we're done. If $n$ is not a prime, then there exists $a$, $b$ such that $n=ab$ with $1<a<n$ and $1<b<n$. Then, by the induction hypothesis, $a=p_1 p_2\cdots p_r$ and $b=q_1 q_2\cdots q_s$, then $n=ab = p_1 p_2\cdots p_r q_1 q_2\cdots q_s$