A Collection of Bogus Proofs

Question

Hello M.S.E. people, This question is just for fun, don't take it seriously :). We have all encountered Bogus Proofs, which seem logical and reasonable, but they prove some claims which are completely wrong and unreasonable. Lets collect a list of very convincing, yet Bogus Proofs. Let me show one down below;

I understand this isn't really a 'question', but its just a fun post to make M.S.E. more interesting and fun.

Answer 1

Here's a classic one; \begin{align*} \frac{\text{d}}{\text{d}x}\left(x^2\right) &=\frac{\text{d}}{\text{d}x}\left(\underbrace{x+x+\cdots+x}_{x\text{ times}}\right)\\ &=\underbrace{\frac{\text{d}}{\text{d}x}\left(x\right)+\frac{\text{d}}{\text{d}x}\left(x\right)+\cdots+\frac{\text{d}}{\text{d}x}\left(x\right)}_{x\text{ times}}\\ &=\underbrace{1+1+\cdots+1}_{x\text{ times}}\\ &=x. \end{align*}

But since $\frac{\text{d}}{\text{d}x}\left(x^2\right)=2x,$ we have; \begin{align*} 2x &=x\\ \implies 2 &=1. \end{align*}