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I often see people who have a solution, and go like "Wow! I proved it!". A solution is a completely different thing than a proof. Here are the definitions of the two from Wolfram Alpha:
Solution | noun | a method for solving a problem.
Proof | noun | a rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition.
See the difference? You didn't prove it, you just showed a solution. Now, I will show you some steps for proof writing.
Step 1: Figure out what you are asked. If the problem said how many steps Jack took to run a mile, you would be looking for the number of steps Jack took. Always state what you are looking for at the beginning of any proof.
Step 2: Prove what you are looking for. There are multiple ways to do this, but I would always recommend a paragraph proof. One type of paragraph proof is induction. This works well when trying to prove an equation. You first prove the equation works for . Then you assume the equation works for . Then you prove it works for . If you prove it works for all of these, the equation is proved. There are many other ways to prove things that I won't go into, but you can find your own ways or look them up.
Step 3: Say what [Step 2] means in terms of the problem. If you use induction, and prove it for those three, so what? How will the persons reading it know what you mean? You need to state what it mean in terms of the problem. If you prove that , what does that mean in the problem?
Step 4: Summarize. You have just proven and explained what everything is. Now it is time to wrap it up. Summarize what you did [Step 2] and write the answer clearly and boxed. Some people might start from the end and go to the front, so write this part clearly. And remember, always end with a QED or an , to show that you are done.
Well, now you know the basics of writing a proof. If you feel I'm missing something, or you just want to say thanks, comment! I'm open to suggestions!