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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Mathematical proofs use logic as a stepping stone to get from your starting point (is this always true?) to your ending point (this is always true/false).
This used logic will depend on what you’re trying to prove, but will typically start with some type of axioms or already known truths (like arithmetic/algebra). Then it’s a matter who making the logical jump from each stepping stone until you’ve reached your ending point. That ending point will either show that your original question is always true or you can have discovered a contradiction which implies your original assumption was incorrect.
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