User blog comment:Fejfo/Weak fixed points/@comment-1605058-20180505100251

A weak fixed point \(\alpha\) of a normal function \(f\) is a fixed point: since \(f(\beta)<\alpha\) for \(\beta<\alpha\), by continuity we must have \(f(\alpha)\leq\alpha\). But since \(f\) is strictly increasing, \(f(\alpha)\geq\alpha\), so \(f(\alpha)=\alpha\).