Can Mathematical Induction Prove Theorems for Negative Numbers?

[kntsy]

Is the principle of mathematical induction unable to prove theorem of negative numbers?

 

[CompuChip]

Why?
If you can prove that if it holds for -n (with n a positive integer), it must hold for -(n + 1), then it is completely equivalent to induction over the positive numbers.

 

[Gib Z]

In fact, the Principle of Mathematical Induction allows us to prove results for Rational Numbers, or any countable set (sets with a bijection to the natural numbers). In practice this can be very hard to do though as it depends on you choosing a "nice" enough bijective function for the particular problem.