I'm confused by https://en.wikipedia.org/wiki/Mathematical_induction#Forward-backward_induction when it says, "Sometimes, it is more convenient to deduct backwards, proving the statement for [MATH]n - 1[/MATH], given its validity for [MATH]n[/MATH]. However, proving the validity of the statement for no single number suffices to establish the base case; instead, one needs to prove the statement for an infinite subset of the natural numbers."
I was trying to find out if mathematical induction can work by (using the ladder analogy) proving an arbitrary rung [MATH]r[/MATH], proving that an arbitrary rung [MATH]k[/MATH] above [MATH]r[/MATH] entails [MATH]k + 1[/MATH], and proving that an arbitrary rung [MATH]j[/MATH] below [MATH]r[/MATH] entails [MATH]j - 1[/MATH]. Basically, sawing the (possibly bidirectionally infinite) ladder into two ladders such that [MATH]r[/MATH] is both the bottom of one and the top of the other. But I don't think that's what it's saying?
I was trying to find out if mathematical induction can work by (using the ladder analogy) proving an arbitrary rung [MATH]r[/MATH], proving that an arbitrary rung [MATH]k[/MATH] above [MATH]r[/MATH] entails [MATH]k + 1[/MATH], and proving that an arbitrary rung [MATH]j[/MATH] below [MATH]r[/MATH] entails [MATH]j - 1[/MATH]. Basically, sawing the (possibly bidirectionally infinite) ladder into two ladders such that [MATH]r[/MATH] is both the bottom of one and the top of the other. But I don't think that's what it's saying?