The previous proof by induction was relatively formal. I want you to use this form initially when I ask for proofs by induction in the first homework. This will help familiarize you with the important parts of such a proof.
However, very few (if any) proofs are completely formal.
A proof is something one human being uses to convince another (intelligent but skeptical) human being that something is true.
Nothing magic about this. The important rule is that, instead of convincing someone by appealing to sympathy or threats, a proof begins with axioms as the basic truths and uses logic as the mechanism for deriving one true statement from another.
