# Claim:Weak induction principle with n-1 is equivalent to weak induction

Suppose you have an inductive proof in the following style:

To prove by weak induction, you can prove and prove for an arbitrary , assuming .

But you are only willing to accept the basic weak induction principle:

To prove "" using weak induction, you must prove (this is often called the base case), and then you must prove for an arbitrary , assuming (this is called the inductive step).

You can systematically convert from the first style to the second:

Suppose you know the following: Then you can conclude , using only the basic weak induction principle.