What are some good subtly incorrect proofs of obviously incorrect results?

Question

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.

1. • $e^{i \pi} = -1$
   • $e^{2i\pi} = 1$
   • $2i\pi = \ln 1$
   • $2i\pi = 0$
   • $-4\pi^2 = 0$

2. Let $a = b$. Then:

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

3. The Two Envelopes Paradox:

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?

Answer 1

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.