I am reviewing some material I learned many years ago about mathematical induction, and I think I may have learned it incorrectly. I have been under the impression that there is a correct and an incorrect way to do certain proofs. For example, assume I want to prove the trivial statement: "for any $z \in \mathbb{Z}_+$, $2z+2$ is even."
One "proof" that I would have written would just acknowledge that, given arbitrary $z \in \mathbb{Z}_+$, $2z +2 = 2\left(z+1\right)$ is even by definition of even numbers. This seems to make sense, and since $z \in \mathbb{Z}_+$ is arbitrary and the proof doesn't depend on choice of $z$, I would say that the conclusion is proven.
I recall my former teachers disagreeing and insisting I use induction, something like:
- $2\cdot 1 = 2$ is even, so the statement holds for $1$.
- Assume $2z + 2$ is even for $z \in \mathbb{Z}_+$...(then show that $2\left(z+1\right) + 2$ is even.)
Why is this? Are both proofs valid? The induction seems to not help much here, but I can see how for some propositions, induction makes proofs simpler.
The reason I am asking this is because the proof of $a^n\cdot a^m = a^{n+m}$ for $n,m\in \mathbb{Z}_+$ is often demonstrated by fixing $n$ to be an arbitrary, then performing induction on $m$, and concluding that since it holds for arbitrary $n$ and all $m \in \mathbb{Z}_+$, it holds for all $n,m\in \mathbb{Z}_+$. I don't understand why induction is "required" for the $m$-variable but not for $n$. Why do I need to do induction at all, and not just say that it is obvious that it holds for arbitrary $n,m \in \mathbb{Z}_+$?
Edit
I should clarify that the rules of exponents example is from Munkres' "Topology", Chapter 1, Section 4, Exercise 6. It is in the section of the book that defines mathematical induction. A hint is provided to prove the formula for fixed $n$ by induction on $m$.
I will also try to be more specific with my question:
- Is my "even-ness" proof without induction valid? Meaning, if I say "$2z+2 = 2\left(z+1\right)$ is even for any $z\in \mathbb{Z}_+$", have I conclusively proved it? I understand I could use induction here, but do I need to?
- If my even-ness proof works, then shouldn't I be able to do the same to prove $a^n \cdot a^m = a^{n+m}$ for $n,m\in \mathbb{Z}$, by simply saying "it is true because it is true for arbitrary fixed $n$ and $m$"? Is the exercise just giving me an excuse to use induction?