The real cofinality of singular cardinals in $L$ under $0^\#$

Question

Suppose that $0^\#$ exists, is there a relatively simple way to show that for any ordinal $\lambda$, if $\lambda$ is a singular cardinal in $L$ then its real cofinality is $\omega$?

Answer 1

The statement is not true. Let $\kappa$ be any regular cardinal in $V$, and let $\lambda=(\aleph_{\kappa+\kappa})^L$. This is a singular cardinal in $L$, because $L$ can see the sequence $\langle\aleph_{\kappa+\alpha}^L\mid\alpha\lt\kappa\rangle$ converging to $\lambda$. But the true cofinality of $\lambda$ in $V$ is $\kappa$, since it is the supremum of an increasing $\kappa$ sequence of ordinals and $\kappa$ is regular.