Why is a certain subset of a regular uncountable cardinal stationary? this is an excerpt from Jech's Set Theory (page 94).

For a regular uncountable cardinal $\kappa$ and a regular $\lambda<\kappa$ let $$E^\kappa_\lambda= \{\alpha<\kappa:\mbox{cf}\ \alpha=\lambda \}$$
  It is easy to see that each $E^\kappa_\lambda$ is a stationary subset of $\kappa$

Well, I don't see so easily why this should hold.
Essentially, given a set which is disjoint to a set of this form, there is no reason for it to be bound, so I tried proving it can not be closed. However, I have no idea why this should hold either, as it might not have any ordinal with cofinality $\lambda$ as a limit point as well.
Thanks in advance

Note that if $C$ is a club, then $C$ is unbounded, and therefore has order type $\kappa$. Since $\lambda<\kappa$ we have some initial segment of $C$ of order type $\lambda$. 
Show that this initial segment cannot have a last element. Its limit is in $C$ and by the regularity of $\lambda$ must have cofinality $\lambda$.
Therefore $C\cap E^\kappa_\lambda\neq\varnothing$.
