The proof of first exit time is a stopping time.

Question

Here is a proof of verifying the hitting time is a stopping time :(the last part of the web page)

https://lecturenotes.math.cmu.edu/mediawiki/index.php/Stochastic_Calculus_(Fall_2012)/Lecture_1

the picture is too small to identify:

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I don't know how to get the "and thus $\sigma\ge\tau$" in the last but one line.

the author probably want to show this conclusion by "$X_\sigma \notin D$".But I don't know how to get "$X_\sigma \notin D$" by "$X_\sigma \notin K_m^\circ$ for arbitrary m". Any suggestions?

Answer 1

By construction,

$$D = \bigcup_{m \geq 0} K_m = \bigcup_{m \geq 0} K_m^{\circ}.$$

Therefore

$$X_{\sigma} \notin D \Leftrightarrow \forall m: X_{\sigma} \notin K_m^{\circ}.$$