TITLE: A Collection of Bogus Proofs
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: 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*}