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Although it is possible to give a precise definition of just what a proof is, we will settle for an intuitive, if less than precise, definition, based on a mathematician's idea of a proof.
A proof is a clear and indisputable argument for the truth of a claim.
The word indisputable needs some clarification.
Our concept of a proof is not a lawyer's concept, where a preponderance of evidence is accepted as "proof". It is not even a scientist's definition, where repeated experiments with the same outcome are considered the closest thing to proofs that they can get. Our definition requires a truly indisputable argument, based on mathematical ideas.
A proof is not some notes that you write to yourself to convince yourself that a claim is true. Rather, a proof is written to convince a skeptic of the truth of the claim. It must be possible for the skeptic to read and check each step, verifying that the proof is correct. You will not be present during the checking, so you cannot rely on additional explanations. The proof stands or falls based on what you have written.
Students have usually grown accustomed to the idea that they provide answers and an instructor checks them. Hence, a student's sole goal is to come up with the correct answer (which the instructor, presumably, knows). But the concept of a proof is intended to be useful even when nobody knows the correct answer. It offers a way for you to find an answer and to convince the world that you are correct.
It can take a lot of thought to discover how to do a proof. You explore different approaches and ideas. But once you have discovered something that works, present that. Do not explain all of the ideas that did not work. You do not even need to explain what inspired you. Just present the proof, making it clear, readable and easy to check.
A common question is how a person knows what to prove. Where do the theorems come from, let alone the proofs? The same principle applies. You try examples that suggest possible theorems. When you think a statement might be true, you try to find a proof, often by looking more closely at examples to understand why the statement should be true. There is a lot of trial and error. But in the end, you should have a clear statement of the theorem and a polished proof.