If $f\colon\kappa\rightarrow \kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded.
This is from Jech's book (page 103) so I guess $\kappa$ is an ordinal or a cardinal. However, it seems to me that if $\kappa$ was an ordinal then the statement would not be true. For example, let $f(0)=f(1)=0$ and for each $n > 1$ let $f(n) = n-1$. Then for each $1 < n < \omega$ we have $f(n)<n$ but $f(\omega) = \omega$. Am I right?
I have found a proof in the internet, but it uses stationary sets and Fodor's Lemma. However, It does seems to me that if I could figure out weather $\kappa$ denotes an ordinal or a cardinal, then I will be able to find a proof for this, which uses more basic definitions, like the definitions of transitive sets an ordinals.
What do you think?