 Solution

Take the Beth numberCantor's theorem makes this sequence strictly increasing. It is a countable cofinal sequence in $\kappa$, while no finite set is cofinal in the limit ordinal $\kappa$. Hence $\mathrm{cf}(\kappa )={\aleph }_0$. ConsequentlyThus Beth omega is a singular strong limit above the continuum. No Continuum hypothesis is needed for the example.