Example 1: A simple arithmetic identity
Consider the identity 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.
Example 3: An example of strong induction
Induction on a general ordered set
Transfinite induction is discussed in a separate article.