Proofs-- Bad Ones

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What bad proofs might go in a course on bad proofs?

I'm thinking such a course could be a fun way to teach students, at grad or undergrad level, how to do proofs. It would include some math classics of the 1=2 and diagonal length of a square equals side length variety with some published economics or math proofs with fatal errors. Answers can be at any level, but the proofs should have one of these features:
(1) Have flaws that are easy to see ex post.
(2) Be short.
(3) Be in a classic paper.
(4) Illustrate a subtle and dangerous flaw.

That is, no sign errors in the middle of 5 pages of algebra in some East Slovakian working paper.