proof of fixed points of normal functions

Suppose $f$ is a $\kappa$-normal function and consider any and define a sequence by ${\alpha }_0=\alpha$ and ${\alpha }_{n+1}=f({\alpha }_n)$. Let . Then, since $f$ is continuous,

So $\mathrm{Fix}(f)$ is unbounded.

Suppose $N$ is a set of fixed points of $f$ with . Then

$$f(supN)=\underset{\alpha \in N}{sup}f(\alpha )=\underset{\alpha \in N}{sup}\alpha =supN$$

so $supN$ is also a fixed point of $f$, and therefore $\mathrm{Fix}(f)$ is closed.