# Proof of the Début theorem

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,

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.