# Is mathematical induction always needed?

When we want to prove that a property P(n) holds for every natural number n, we can, and must use mathematical induction. So I was wondering if it is wrong if we DON'T use induction in obvious mathematical statements. For example, let's solve this exercise below, without mathematical induction.

Exercise: Prove that $$n^2-1=(n-1)(n+1)$$ for every $$n\in \mathbb{N}.$$

Solution: Suppose $$n\in \mathbb{N}$$. Then $$(n-1)(n+1)=n\cdot n + n\cdot 1 -1 \cdot n - 1\cdot 1=n^2-1$$.

Since $$n\in \mathbb{N}$$ was arbitary, $$n^2-1=(n-1)(n+1)$$ holds for every $$n\in \mathbb{N}.$$

So is the above solution correct? Are we obliged to use only mathematical induction to prove that such a statement holds for every $$n \in \mathbb{N}$$?

• You are answering yourself by exhibiting a valid proof that doesn't use induction.
– user65203
Sep 21, 2018 at 8:59
• @YvesDaoust Daoust This response is so cool! :p Thank you for the answer! Sep 21, 2018 at 20:31