How to prove the following statement regarding the successor function and addition of natural numbers?

Question

Natural numbers (including 0) and the successor function are defined as per the Peano Axioms (you can check them on wikipedia). Addition is defined recursively as follows:

$a+0=a$

$a+S(b)=S(a)+b$

With these definitions in mind, I need to prove that $a+S(b)=S(a+b)$

I tried using induction on b, and while proving the base case was trivial, I only end up with a circular argument when I try to prove the inductive step.

Answer 1

Hint: try to prove the stronger claim: $$ \forall x(a + S(x) = S(a+x)) $$