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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 $n \in \mathbb{N}$?

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

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You don't HAVE to use induction when working on natural numbers. You MAY use it, and you SHOULD at least give it a try if nothing obvious appears. It's a really powerfull tool, but some cases, just like the example in your question can be solved directly without having to rely on induction

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  • $\begingroup$ My bad, it's exEmple in French, I'll edit that. Thanks! $\endgroup$
    – F.Carette
    Sep 21, 2018 at 9:05
  • $\begingroup$ Pardon mon ami, I don't know much French. $\endgroup$ Sep 21, 2018 at 9:08
  • $\begingroup$ @F.Carette That was very helpful. Thank you! $\endgroup$
    – kafroulis
    Sep 21, 2018 at 20:29

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