What are some classic fallacious proofs?
If you know it, also try to include the precise reason why the proof is fallacious. To start this off, let me post the one that most people know already:
1. Let $a = b$.
2. Then $a^2 = ab$
3. $a^2 - b^2 = ab - b^2$
4. Factor to $(a-b)(a+b) = b(a-b)$
5. Then divide out $(a-b)$ to get $a+b = b$
6. Since $a = b$, then $b+b = b$
7. Therefore $2b = b$
8. Reduce to $2 = 1$
As @jan-gorzny pointed out, in this case, line 5 is wrong since $a = b$ implies $a-b = 0$, and so you can't divide out $(a-b)$.