  We search for the $\psi(4S)$ state in $B^\pm \ra \eta J/\psi K^\pm$ and
$e^+e^- \ra \eta J/\psi$ based on the Belle measurements with the assumed mass
$M=(4230\pm8)$ MeV/$c^2$ and width $\Gamma=(38\pm12)$ MeV. No significant
signal is observed in the $\eta J/\psi$ mass spectra. The 90\% confidence level
upper limit on the product branching fraction $\BR(B^\pm \ra
\psi(4S)K^\pm)\BR(\psi(4S) \ra \eta J/\psi)<6.8\times 10^{-6}$ is obtained in
$B^\pm \ra \eta J/\psi K^\pm$ decays. By assuming the partial width of
$\psi(4S) \to e^+e^-$ being 0.63 keV, the branching fraction limit
$\BR(\psi(4S) \ra \eta J/\psi) < 1.3\%$ is obtained at the 90\% confidence
level in $e^+e^- \ra \eta J/\psi$, which is consistent with the theoretical
prediction.
