# Proof: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.
Proof: By changing variable names
Assume the statements that are given in the claim; we will show and for an arbitrary , assuming .

is easy: it was given to us as assumption 1.

To see , choose an arbitrary , and assume . Let . Then , and we have assumed . Thus we can apply assumption 2 given above to conclude .

But , which is what we wanted to prove.