Tricki

## Revision of Induction front page from Fri, 24/04/2009 - 20:37

 This article is a stub.This means that it cannot be considered to contain or lead to any mathematically interesting information.
 Incomplete This article is incomplete. See comments.

### Example 1: A simple arithmetic identity

Consider the identity

Since the left-hand side of (1) is neither an arithmetic nor a geometric progression, we are unable to use the standard formulas for calculating such sums; and, indeed, it is not obvious how to transform the expression into one that can be manipulated into a formula for the sum. However, this can be proved using induction.

### See also

Strengthen your inductive hypothesis Transfinite induction is discussed in a separate article.

## Comments

### Moved the discussion to the comments

When this article is written, it should have examples of several different styles of inductive proof: the usual kind where you deduce from , the slightly more sophisticated kind where you deduce it from the fact that is true for every , induction over more sophisticated well-ordered sets, use of the well-ordering principle, etc.

### Induction without an indexing set

"The generalized induction is harder because we don't have a nice indexing set like the natural numbers. Is there a way of doing induction without the crutch of an indexing set? Yes there is: you look for a minimal counterexample."

The above is from this article by Prof Gowers: http://www.dpmms.cam.ac.uk/~wtg10/bounded.html.

I feel that this could be put somewhere, but I am not sure where. The connection between the well-ordering principle and induction could also be mentioned (albeit briefly, since it doesn't seem like a trick).