Question

Principle of inclusion and exclusion proof by induction

Answer 1

You start by writing the full statement that you need to prove. Next you isolate the variable that you intend to do the induction. Then you write down the base case for $n=0$ or $n=1$, you continue by writing the complete statement for the induction hypothesis for $n$ and then you proceed to prove the case for $n+1$. It is often the case that we forget to write the statement that we want to prove, and then we don't know what we need to prove. Writing it down helps us understand it better, and sometimes it even simplifies everything to a purely technical (and sometimes simple too) manipulation.