Here, $(\kappa)^+$ is the Hartog's Number, the least cardinal $\lambda$ so that there is no injection from $\lambda$ to $\kappa$.
My feeling is that this is true, since for any ordinal $\alpha < \omega $, $\ \alpha+1<\omega$, so for each infinite cardinal $\aleph_\alpha<\aleph_\omega$, $\ \aleph_{\alpha+1}=(\aleph_{\alpha})^+<\aleph_\omega$, but I feel I am making some leaps in logic here or have missed something and this doesn't constitute any kind of a proof.