I was reading principle of recursive definition from Toplogy by James R. munkres. And there i saw a this statement that I can't understand
recursion is something that you define by induction but not prove by induction. My question is what is the definition of induction then??
I would like to summarise the process of induction as two things:
1) The process of induction involves checking the base case for $n$, setting $n = k+1$, and rearranging the expression to prove the case also works for $k+1$.
2) Induction is specifically used for integers only. It is strong for proofs only involving integers, as you can repetitively define $k+1$ as $k$ to get $k+2, k+3...$ and so on. However, many problems in all fields of algebra (in my experience) use rational numbers (that are fractions), irrational numbers, complex numbers, and so is a "weaker" method of proof.