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I was reading the stochastic calculus notes on this website and I read the following in the proof of the Début theorem but I could not understand what does it mean. Can someone explain it to me rigorously? Why is this statement true?

http://almostsure.wordpress.com/2009/11/15/stopping-times-and-the-debut-theorem/

Continuous processes always achieve their supremum value on any compact interval,

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We deal with continuous stochastic processes quite often (e.g. Brownian motion). The term continuous in here refers to having continuos paths, that is, a stochastic process $X=(X_t)$ is such that for fixed $\omega \in \Omega$ (where $(\Omega, \mathcal{F}, P)$ is the corresponding probability space) a function (a path of $X_t$) $$ t \mapsto X_t(\omega)$$ is continuous.

Bolzano–Weierstrass theorem implies that every continuous function on a compact interval attains its bounds (supremum and infimum).

Therefore, the statement which you were not sure about holds via Bolzano–Weierstrass theorem. For more details you can also see Prove that a continuous function on a closed interval attains a maximum.

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  • $\begingroup$ thanks you for clearing it up for me. I guess I need to review some basic real analysis as I have forgotten a few results $\endgroup$ – user3503589 Oct 27 '14 at 15:17

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