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Normally, when using induction, I assume a statement is true for n, then I will try to show the same statement is also true for n+1.

In the problem I have now, is is correct if I assume a statement is true for n+1, then show that the statement is true for n, the the whole statement is true.

Please give the insight.

Peter
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    You really ought to state the problem or else there's no way of knowing whether you're doing something correct. As a general principle: if you know that a proposition $P(n)$ is true for, say, $n = N$ for some large $N$, then backwards induction starting at $N$ works. Otherwise it doesn't. – Ryan Reich Nov 07 '13 at 23:09
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    No. Let $P(n)$ mean $n\leq 10$. You can never get $\forall n\in \Bbb NP(n)$ – Git Gud Nov 07 '13 at 23:10

1 Answers1

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The reason why induction works is because you prove the base case.

It works as follows, let $x_n$ represent that statement "Our theorem is true for $n$".

$$x_0 \Rightarrow x_1 \Rightarrow x_2 \Rightarrow \cdots $$

and since we know $x_0$ is true we can just work up this path to conclude it is true for all $n$.

In your case you want to work backwards. To do this you need somewhere to start. For instance if you know the statement is true for $17$, and you have proved "reverse induction" you can conclude it is true for all $n\le 17$ as follows:

$$x_{17} \Rightarrow x_{16} \Rightarrow x_{15} \Rightarrow \cdots \Rightarrow x_{0}$$

But you see you have to have somewhere to start to work your way down this trail. So this method will never imply something is true for all natural numbers.

Deven Ware
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