TITLE: "Too simple to be true"
QUESTION [32 upvotes]: 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...

REPLY [2 votes]: There is simple "proof" of four color theorem:
http://www.superliminal.com/4color/4color.htm
unfortunately I still can't see the gap in it.