Is it true that given stationary set $A$ of $k$, there exists an ordinal $\alpha < k$ such that $A$ contains all limit ordinals of $k$ greater then $\alpha$?
I think it is true because the set of all limit ordinals of $k$ is stationary (i think..)
Thanks
No, it's not true at all.
First of all, the set of limit ordinals below $\kappa$ is a closed and unbounded set (at least assuming $\operatorname{cf}(\kappa)>\omega$). This means that it is much harder to include a tail segment of the limit ordinals, since it means including a closed and unbounded set.
For example, consider $\kappa=\omega_2$, then the set $\{\alpha<\omega_2\mid\operatorname{cf}(\alpha)=\omega\}$ is a closed and unbounded set, but $\omega_1$ is not there, and not any ordinal of the form $\alpha+\omega_1$ as well. So it certainly doesn't include all the limit ordinals above a certain point.
What is true, however, is that if $S$ is stationary then it has arbitrarily large limit ordinals below $\kappa$.