It is helpful as you learn to read and write mathematical proofs that you familiarize yourself with several common "methods" of proof. As you look over these examples, do not concentrate on formality but on intent. For example, because you will be familiar with the general form of a proof by contradiction, you will more easily understand such proofs since you will know what to expect next. A good writer will signal to you that he intends to prove his claim by contradiction by starting the proof with something like "Suppose not." After reading this, you know you should look out for the contradiction. Once the contradiction has been established, you should be satisfied that the claim has been proven.
If you and the author have never heard of "proof by contradiction", it is still possible to communicate the same proof. It is just easier to communicate when both author and reader follow an established format. For example, try telling a knock-knock joke to someone, perhaps a small child, who has never heard of knock-knock jokes. You'll find it very difficult to deliver the punch line.
You should also realize that these are not methods for discovering proofs, but for writing proofs. Before you even begin writing a proof, you should already have a good idea why it is true. The writing up the proof helps you check that your "good idea" is in fact correct and allows you to communicate your idea to others.
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