Say I have a statement $P(n)$ which is true for all natural numbers $1 \leq n \leq 100$, but is nonsensical or false for other values of $n$. If I want to prove that $P(n)$ is true for $1 \leq n \leq 100$, and it seems induction is the best option, I'll often use induction to prove the following statement:
"For all $n\in\mathbb{N}$, If $n \leq 100$, then $P(n)$."
I have two concerns with this:
- It adds awkward and tedious wording to the proof. For example, in the induction step, I'll say something like this: "Suppose for some $k \in \mathbb{N}$ that if $k \leq 100$, then $P(k)$. Now suppose $k+1 \leq 100$. It follows that $k \leq 100$, so we have $P(k)$..."
- Sometimes $P(101)$ is not simply false, but undefined, so the bolded statement of the second paragraph is meaningless. I feel weird including a meaningless statement in a proof.
Am I right to be troubled by this? If so, is there any phrasing I can use to accomplish such a proof while avoiding these issues?
You can just say, "Assume $n<100$ and that $P(n)$ is true." Then you prove $P(n+1).$ You don't need any conditions on $n$ in the induction hypothesis.