TITLE: Examples of interesting false proofs
QUESTION [69 upvotes]: According to Wikipedia False proof

For example the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way.

The Wikipedia page gives examples of proofs along the lines $2=1$ and the primary source appears the book Maxwell, E. A. (1959), Fallacies in mathematics.

What are some examples of interesting false proofs?

REPLY [5 votes]: This proof that $\pi=0$ may be of some interest in examinations.
The function $f(x)=\arctan(x)+\arctan(1/x)$ has derivative $f’(x)=\frac1{1+x^2}-\frac1{x^2} \frac{1}{1+\frac1{x^2}}=0$, hence it is constant. Therefore$\displaystyle \lim_{x\to+\infty}f(x)= \displaystyle \lim_{x\to-\infty}f(x)$, that is $\frac\pi2=-\frac\pi2$, whence $\pi=0$.$\quad\square$