I miss one step in understanding the proof of the Debut theorem for right-continuous adapted processes as it is given here. This is my problem. Given a process $X$ consider the random time $$ S = \inf\left\{t\geq 0 \left| \sup_{0\leq u\leq t}X_u\geq K\right.\right\}, $$ where $K$ is a real constant. The proof says that the event $\{S\leq t\}$ is equivalent to the event $\{\forall\varepsilon>0\exists 0\leq s\leq t|X_s>K-\varepsilon\}$. I am really struggling in understanding why so any advice is warmly welcome.
ADDENDUM It is probably more intuitive this formulation of the problem. Let $f$ be a function and let
$$x_0=\inf\left\{y\geq 0\left|\sup_{0\leq z\leq y}f(z)\geq K\right.\right\}.$$
Therefore
$$x_0\leq x\Leftrightarrow \forall\varepsilon>0\exists x^{\prime}\in[0,x]:f(x^{\prime})>K-\varepsilon$$