Let $\kappa$ be a singular cardinal, such that $\operatorname{cf}\kappa = \lambda < \kappa$. Now because $\operatorname{cf}\kappa = \lambda$, then I can write down an increasing sequence of ordinals $\langle \alpha_{\xi} \mid \xi < \lambda \rangle$ such that $\displaystyle\lim_{\xi \rightarrow \lambda}\alpha_{\xi}=\kappa$.
But why is it possible to construct an increasing, continuous sequence of cardinals $\langle \beta_{\xi} \mid \xi < \lambda \rangle$?
I think that replacing each $\alpha_{\xi}$ with $| \alpha_{\xi}|^+$ yields an increasing sequence of cardinals (since $\kappa$ is not a successor cardinal), but how do I guarantee continuity? Do we require that $\lambda > \omega ?$
Any help would be appreciated.