Here is a faulty proof of the circular law of random matrix theory:
Take a random matrix $M$ with i.i.d. complex entries and write it as $A+iB$, where $A:=(M+M^*)/2$ and $iB:=(M-M^*)/2$ are the Hermitian and skew-Hermitian parts of $M$, respectively. By the Wigner semicircle law, a randomly picked eigenvalue of $A$ or $B$ obeys the Wigner semicircle distribution, that is, the distribution of the first component of a random point on the disk. Hence, a randomly picked eigenvalue of $A+iB$ is uniformly distributed in the disk.
Obviously the problem with this proof is that eigenvalues are not additive -- the $k$th eigenvalue of $M$ is not the $k$th eigenvalue of $A$ plus the $k$th eigenvalue of $iB$.
Nevertheless, is there any moral truth to the proof above, maybe because the randomness makes eigenvalues almost additive in the required sense?