In principle, a proof can be any sequence of logical deductions from axioms and previously-proved statements that concludes with the proposition in question. This freedom in constructing a proof can seem overwhelming at ﬁrst. How do you even start a proof? Here’s the good news: many proofs follow one of a handful of standard templates. Proofs all diﬀer in the details, of course, but these templates at least provide you with an outline to ﬁll in. We’ll go through several of these standard patterns, pointing out the basic idea and common pitfalls and giving some examples. Many of these templates ﬁt together; one may give you a top-level outline while others help you at the next level of detail. And we’ll show you other, more sophisticated proof techniques later on. The recipes below are very speciﬁc at times, telling you exactly which words to write down on your piece of paper. You’re certainly free to say things your own way instead; we’re just giving you something you could say so that you’re never at a complete loss.
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