I understand how the steps in a proof are inferred from other steps that are either given or already inferred. Sometimes though, in previous lectures, a professor would begin proving a certain theorem or equation (such as in calculus and statistics) and then at some point that may as well have been arbitrary to me, declared it proven. What decides the last step in a proof and what about it is so special that it “proves” the subject?
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Every step in the proof prove something. We stop when we reach the claim we originally set out to prove, but all claims that had been proved in the proof without additional assumptions are logically valid.
There are no special rule about which claim is the one we set out to prove. It’s not unusual to not have goal at first, and see what you can get, which happens a lot at research level. But when you’re in class, there is always a goal, because of 2 reasons. One, the topic is already understood, so people know which claims is important and useful, and that’s the one being taught to you. Two, if you’re being tested, then you have to be given a goal to reach.
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