What are some good subtly incorrect proofs of obviously incorrect results? [duplicate]

I'm interested in compiling a list of proofs that look logically correct at a glance, but "prove" something obviously incorrect. Here are some examples.

• $e^{i \pi} = -1$
• $e^{2i\pi} = 1$
• $2i\pi = \ln 1$
• $2i\pi = 0$
• $-4\pi^2 = 0$
1. Let $a = b$. Then:

• $a = b$
• $a^2 = ab$
• $a^2 - b^2 = ab - b^2$
• $(a+b)(a-b) = b(a-b)$
• $a + b = b$
• $a = 0$

You are on a game show, in which the host fills two indistinguishable envelopes with random amounts of money, such that one envelope contains $x$ dollars and the other contains $10x$ dollars. You pick an envelope at random, but then you are offered a chance to switch envelopes (intuitively, it shouldn't matter whether or not you choose to switch). You reason: there is a 50/50 chance that I currently hold the higher-valued or lower-valued envelope in my hands. If I keep this envelope, my expected return is $x$. If I switch, then my expected return is $\frac{1}{2}(\frac{1}{10}x) + \frac{1}{2}(10x) = 5.05x$. Therefore, I should switch.

Any other good ones?

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marked as duplicate by Bill Cook, Potato, Goos, Andrey Rekalo, MicahSep 13 '13 at 5:40

Also: should I have made this a community wiki post, since it asks for a big list but doesn't have a single definite "acceptable" answer? –  GMB Sep 13 '13 at 3:56
Also, this thread is a possible repeat, but it didn't gain too much traction. –  GMB Sep 13 '13 at 3:57
Yes, you should have made it CW. I don't think the first two are examples of "subtly incorrect" proofs, because the basic error is not subtle at all. Both start from $f(x) = f(y)$ for some noninjective $f$, and then conclude $x=y$, and there are probably dozens of examples of this error in the folklore and on this web site. –  MJD Sep 13 '13 at 4:00
Also if you are going to claim that your question is different from What are some classic fallacious proofs?, you should find at least one example that is not given there. –  MJD Sep 13 '13 at 4:02
@MJD: I didn't find the repeat thread until after I posted this one, so the repeat examples are a coincidence (maybe there just aren't that many good fallacious proofs floating around). But either way, #1 is not a repeat. In terms of "too subtle," I planned to use these on advanced-math high schoolers, not career mathematicians. Treat the subtlety of the examples as a guideline for Minimum Required Subtlety. –  GMB Sep 13 '13 at 4:06

Well, here's one mentioned by the logician/mathematician Timothy Chow. It's a "proof" that $0 = 1$ using something more interesting than the usual division-by-zero tricks; specifically it's a "proof" via advanced calculus, involving a differentiation under the integral sign.