proof of fixed points of normal functions

Suppose $f$ is a $\\kappa$ -normal function and consider any $\\alpha<\\kappa$ and define a sequence by $\\alpha_{0}=\\alpha$ and $\\alpha_{n+1}=f(\\alpha_{n})$ . Let $\\alpha_{\\omega}=\\sup_{n<\\omega}\\alpha_{n}$ . Then, since $f$ is continuous,

f(\\alpha_{\\omega})=\\sup_{n<\\omega}f(\\alpha_{n})=\\sup_{n<\\omega}\\alpha_{n+1}=%\\alpha_{\\omega}

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

Suppose $N$ is a set of fixed points of $f$ with $|N|<\\kappa$ . Then

f(\\sup N)=\\sup_{\\alpha\\in N}f(\\alpha)=\\sup_{\\alpha\\in N}\\alpha=\\sup N

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