  Let $U_n$ be an $n \times n$ Haar unitary matrix. In this paper, the
asymptotic normality and independence of $\Tr U_n, \Tr U_n^2, ..., \Tr U_n^k$
are shown by using elementary methods. More generally, it is shown that the
renormalized truncated Haar unitaries converge to a Gaussian random matrix in
distribution.
