I've always enjoyed false proofs, but the only one I saw on here was an algebraic one. Here's one I stubled upon way back when I was learning the Calculus (do people mind TeX notation? It shouldn't be too cryptic.)
To evaluate the integral of 1/x, we use (the much loved) integration by parts. Recall that integration by parts exploits the following:
\int u dv = uv - \int v du.
Let u = 1/x, then du = -1/x^2 dx. Let v = x; therefore, dv = dx. We write
\int 1/x dx = \int u dv
\int 1/x dx = uv - \int v du
\int 1/x dx = (1/x)x - \int x (-1/x^2) dx
\int 1/x dx = 1 + \int 1/x dx.
Subtracting the original integral from both sides leaves 0 = 1. QED Note, this solves the pesky problem of having to ever evaluate (or even define) the log base e (an irrationally based logarithim, if ever one was!).
Anyone seen that before? I haven't.