I'm struggling with the proof of the upcrossing inequality for stochastic processes:
Let $(X_t)_{t \geq 0}$ be a submartingale w.r.t. a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, P)$ such that for every $\omega \in \Omega$ the map $t \mapsto X_t(\omega)$ is continuous. Let $[\sigma, \tau]$ be a subinterval of $[0, \infty)$ and let $\alpha < \beta, \lambda > 0$ be real numbers. Then we have $$\mathbb{E}[U_{[\sigma, \tau]}(\alpha, \beta, X(\omega))] \leq \frac{\mathbb{E}[X_\tau^+] + |\alpha|}{\beta - \alpha}$$ where $U$ denotes the number of upcrossings of $X$ from $\alpha$ to $\beta$ on the interval $[\sigma, \tau]$.
This version of the upcrossing inequality and the proof I am concerned with is taken from Theorem 6 of these lecture notes.
What I don't understand is the proof of "$\leq$" in the proof of claim 2. Since $F$ may also contain non rational numbers, why must $U_F \leq U_{F_k}$? Or, to put it more generally: if we try to prove the upcrossing inequality by using the discrete time result and finding an increasing series of sets of rational indices that converges towards a set thats dense in the interval $[\sigma, \tau]$, how can we ensure that we have upcrossings that happen within non rational numbers?
Alternatively, I'd also be very happy to see an alternative rigorous proof of the continuous time upcrossing inequality.
Thanks!