### Weak induction

### 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.### Example 2: The base case is important

Sometimes one can have the temptation of looking at the base case of induction as a mere formal step that is always trivially true, thinking that the "big deal" is just on the induction hypothesis. But actually it IS important: here we have an example of a general statement which induction hypothesis is true but that is *always* false... because we can never find a base case!

Suppose true that, for a fixed , . Then, . But we obviously know that this statement is never true, so there can't be a base case.

### Strong induction

### Example 3: An example of strong induction

### Induction on a general ordered set

### See also

Strengthen your inductive hypothesis

Transfinite induction is discussed in a separate article.

See also

Using generators and closure properties

A non-trivial circular argument can often be usefully perturbed to a non-circular one.

## Comments

## Moved the discussion to the comments

Fri, 24/04/2009 - 19:08 — brownhWhen 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

Sat, 23/05/2009 - 14:37 — Jungle"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).