Suppose we are trying to prove some $P(n)$ by using proof by induction
First we show that $P(0)$ is true
Then we assume that $P(n-1)$ is true
In practically every math proof I have seen, in the inductive step we start with $P(n)$ and break it into $P(n-1)$ plus something else to show $P(n)$ is true
My question is, can we work forwards? i.e., in my inductive step can I say that it is true for $P(n-1)$, then we "build something up" to show that this $P(n-1)$ plus "something else" is true and that this is really $P(n)$?
If that doesn't make sense, here is a specific example:
i.e. if I am trying to prove some property of a graph in my inductive step, can I start from some $n-1$ verticy graph (my hypothesis) and say that we can add nodes/edges to make it a graph with $n$ verticies, then prove that this graph we can build satisfies the proof?
