I'm reading the Proof that $\Diamond$ implies the existence of a Suslin tree in Jech, Set Theory (2003), p.241. The nodes in the constructed tree are countable ordinals, so $T=\omega_1$, and every subtree $T_{\alpha}$ is a initial segment of $\omega_1$.
Here is a screenshot of the construction of $T$:
Later on, the author states, that it "follwos easily from the construction that for a closed unbounded set of $\alpha$'s, $T_{\alpha}=\alpha$." I can't see why this follows.
Thank you for your help!
Since $T_\alpha$ is an initial segment of $\omega_1$, the function $\alpha\mapsto\sup T_\alpha$ is an increasing function and at limit points $T_\delta=\bigcup_{\alpha<\delta}T_\alpha$. So the function is normal.
And every normal function has a closed and unbounded set of fixed points, namely $\alpha$ such that $T_\alpha=\alpha$.