I was asked to prove the following.
Prove that if $n$ is a natural number greater than $1$, then $n-1$ is also a natural number.
I was also given the following hint.
Prove that the set $\{n|n=1 \text { or }n\in\mathbb{N}\text{ and }n-1\in\mathbb{N}\}$ is inductive.
I am struggling with this on two fronts.
First, I am unsure how to precisely prove that this set is inductive in the first case. Obviously, it includes $1$, but how do I argue that, if it includes $n$, that it includes $n+1$?
Second, I apparently don't fully understand induction, because even given that the hint is true, I don't see how formulating the question like that satisfies the given question.
Any clarification is greatly appreciated.
Well, first we will prove. Obviosly, $1\in X=\{n|n=1$ or $n\in\mathbb{N}$ and $n-1\in\mathbb{N}\}$. Suppose $n\in X$, but $n\neq 1$. So $n\in\mathbb{N}$ and $n-1\in\mathbb{N}$. Let $m=n+1$. So $m-1\in\mathbb{N}\Rightarrow m-1\in X\Rightarrow m=n+1\in\mathbb{N}$.
When you prove $n\in X\Rightarrow n+1\in X$, you prove that if $n+1\in\mathbb{N}$ with $n+1\neq 1\Rightarrow n\in X$, so $n=1$ or $n\in\mathbb{N}$ and $n-1\in\mathbb{N}$, so $n\in\mathbb{N}$.