As indicates the title, this question is about "proofs" of true statements which are short and/or look elegant but are wrong.
I mean example like Cayley-Hamilton's theorem, which states that for a $n\times n$ matrix over $\Bbb C$, and $\chi$ its characteristic polynomial, then $\chi(A)=0$. The well-known fake proof consists of a substitution $\lambda=A$ in $\chi(\lambda)=\det(A-\lambda I)$, which is not allowed.
So, I think writing a big-list could be interesting, where each answer will contain:
- the statement;
- the fake proof;
- an explanation of the gap in the proof;
- if possible, a reference to a good proof.
Each one can concern any field of mathematics. It will be good to have an example in every field: real analysis, measure theory, etc...