Is it true that for any cardinal $\kappa$ that in injects into $\aleph_{\omega}$, $(\kappa)^{+}$ injects into $\aleph_{\omega}$?

Question

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.

Answer 1

It is true that $\aleph_\alpha<\aleph_\omega\implies\aleph_{\alpha+1}<\aleph_\omega$ - this is true simply because $\omega$ is a limit ordinal - but "$\aleph_\alpha$ injects into $\aleph_\omega$" does not imply $\aleph_\alpha<\aleph_\omega$ (e.g. $\aleph_\omega$ injects into $\aleph_\omega$, obviously).