How $\tau$ is a limit point of $\overline{D \cap(-\infty, t)}$?

Question

I'm reading a lemma about monotone functions in textbook Analysis I by Amann.

Here is Prop 2.9:

To apply Prop 2.9, it is required that $\tau$ is a limit point of $\overline{D \cap(-\infty, t)}$, but I'm unable to understand.

Could you please elaborate on how $\tau$ is a limit point of $\overline{D \cap(-\infty, t)}$? Thank you so much!

Answer 1

I do think they messed up the order of the proof in the book a little.

Suppose that $\overline{D_{t}}\neq\emptyset$ and $\tau\in\overline{D_{t}}$. In particular we have that $$\tau\in\overline{(-\infty,t)}\cap\overline{(t,\infty)}=[-\infty,t]\cap[t,\infty]=\{t\},$$ so $\tau=t$. Note that $t\not\in D\cap(-\infty,t)$ and $t\not\in D\cap(t,\infty)$, so if $t\in\overline{D_{t}}$ we have that $t$ is a limit point of $D\cap(-\infty,t)$ and $D\cap(t,\infty)$.