Here’s a nice guide to what is a proof and how to write one:
How to write proofs: a quick guide by Eugenia Cheng.
This is brief, has a nice outline, and gives some good examples. I love the analogy made, that a good proof is like a good story: it has a beginning, middle and end. The point: a proof can be bad by having the parts in the wrong order.
A shortcoming of the article, in my opinion, is the explanation of an induction proof. It’s a pet-peeve of mine, but I dislike seeing “n = k” and “n = k+1” in the same paragraph. Problems with teaching and learning induction is a greater issue that is partially addressed in the May/June 2008 article Some Observations on Teaching Induction by Mary E. Flahive and John W. Lee.