  The magnetically charged SU(2) Reissner-Nordstr\"om black-hole solutions of
the coupled nonlinear Einstein-Yang-Mills field equations are known to be
characterized by infinite spectra of unstable (imaginary) resonances
$\{\omega_n(r_+,r_-)\}_{n=0}^{n=\infty}$ (here $r_{\pm}$ are the black-hole
horizon radii). Based on direct {\it numerical} computations of the black-hole
instability spectra, it has recently been observed that the excited instability
eigenvalues of the magnetically charged black holes exhibit a simple universal
behavior. In particular, it was shown that the numerically computed instability
eigenvalues of the magnetically charged black holes are characterized by the
small frequency universal relation $\omega_n(r_+-r_-)=\lambda_n$, where
$\{\lambda_n\}$ are dimensionless constants which are independent of the
black-hole parameters. In the present paper we study analytically the
instability spectra of the magnetically charged SU(2) Reissner-Nordstr\"om
black holes. In particular, we provide a rigorous {\it analytical} proof for
the {\it numerically}-suggested universal behavior
$\omega_n(r_+-r_-)=\lambda_n$ in the small frequency $\omega_n r_+\ll
(r_+-r_-)/r_+$ regime. Interestingly, it is shown that the excited black-hole
resonances are characterized by the simple universal relation
$\omega_{n+1}/\omega_n=e^{-2\pi/\sqrt{3}}$. Finally, we confirm our analytical
results for the black-hole instability spectra with numerical computations.
