Prove that set is stationary.

Question

In Thomas Jech's 'Set Theory' there is the following statement: For a regular uncountable cardinal κ and a regular λ < κ the set $\{\alpha < \kappa: cf(\alpha) = \lambda\}$ is stationary. But how one could prove it?

Answer 1

Suppose that $C$ is a club. Write it as $\{c_\alpha\mid\alpha<\kappa\}$ in its order. What can you say about the cofinality of $c_\lambda$?