Why doesn't mathematical induction work backwards or with increments other than 1?
Solution 1:
It does work in both ways that you suggested. Typically induction problems are stated so that you want to prove $P(n)$ for all natural numbers $n$. Then you start with $P(1)$ and go from $n$ to $n+1$. However, if you wanted to prove it for negative integers instead you would start with $P(-1)$ and go from $n$ to $n-1$. Of course, you can just change the variable $m:=-n$ and reduce the second case to the first. The same goes for your other example with $m:=10(n-1)$. Since 'backwards' and 'fractional' induction reduce to the standard induction they are equally justified.
Solution 2:
I'm going to be brave and disagree with most of the other answers. (Exception: I agree with, and have upvoted, Dan Piponi's answer; but I think that that answer buries the lede.)
You've been given a theorem in class, along these lines:
Let $P(n)$ be a statement about $n$, with these properties:
- $P(1)$ is true.
- For all $k \in \mathbb{N}$, if $P(k)$ is true then $P(k+1)$ is true.
Then $P(n)$ is true for all $n \in \mathbb{N}$.
and your teacher wants you to write proofs using that theorem.
Using your mathematical intuition, you have correctly divined that there are many other, similar theorems, such as one theorem to cover "all integers $m \le 1$" by starting with $1$ and counting backward, and one theorem to cover "all numbers of the form $1+\frac{p}{10}, p \ge 0$". But those theorems aren't "givens"; they don't appear in your textbook, and your teacher hasn't taught them to you. When you're asked to write a proof for class, it is a tacit requirement that your proof depend only on a certain set of preexisting results that it's O.K. for you to use — the axioms and theorems you're given in class, plus (probably) mechanical algebra and arithmetic. Without that tacit requirement, you could "prove" everything by simply stating that it's a correct statement and therefore proven. So I think your teacher is right to expect you to use the canonical formulation of induction you've been given in class, rather than other similar theorems.
That said, your modified theorems can be proven fairly straightforwardly as corollaries of regular mathematical induction. Dan Piponi's answer shows how to do that for one of your theorems. So you can use your modified theorems, but only as an intermediate step: first you prove the modified theorem, then you use it to prove what you are actually asked to prove. But this is probably unnecessary busy-work; you might as well just combine the two steps, and write your proof directly in terms of the theorem you've been given.