TITLE: When do we use hidden induction?
QUESTION [22 upvotes]: If I'm correct, hidden induction is when we use something along the lines of "etc..." in a proof by induction. Are there any examples of when this would be appropriate (or when it's not appropriate but used anyway)?

REPLY [4 votes]: Hidden induction happens a lot in cases where you go backwards from $n$ to $1$, using some kind of reduction argument. For example, the proof that every number can be written as a product of primes:

Let $n$ be some number. If it's prime, then we're done. Otherwise it can be written as $ab$, with $a, b < n$. Again, if both $a$ or $b$ are prime, we're done, otherwise they can be broken up in the same way. Since the factors are getting smaller and smaller, this process must stop eventually, but the only way it can stop is if one of all of the numbers involved are prime.

Induction serves to tighten up the structure of the argument, replacing a vague "this process must stop" with an explicit invocation of an axiom:

Suppose every number less than $n$ can be written as a product of primes. If $n$ is prime, we're done. Otherwise, $n=ab$ with $a, b < n$, so $a$ and $b$ can be written as products of primes. Therefore $n$ can be written as a product of primes.