Left continuous adapted process X is optional

Question

I was reading through "Limit Theorems for Stochastic Processes" by Jacod and Shiryaev and came across

1.24 Proposition: Every process $(X_t)_{t \geq 0}$ that is left continuous and adapted is optional.

The setup is the follwoning: Let $(\Omega,\mathcal{F},(F_t)_{t \geq 0}, P)$ be a filtered probability space with right continuous filtration.

In the proof, they define a new sequence of process $$X^n := \sum_{k \in \mathbb{N}}X_{k/2^n} 1_{[[k/2^n,(k+1)/2^n[[}$$ and claim that for every $(t,\omega) \in [0,\infty) \times \Omega$, we have $X^n_t(\omega) \rightarrow X_t(\omega)$ for $n \rightarrow \infty$. Why does $X^n$ converge pointwise to $X$ ? I do not see why this has to hold.

Thanks a lot in advance!

Answer 1

It's pretty simple: For $t \geq 0$, we have $$ X_t^n = X_{2^{-n}\lfloor t2^n \rfloor}. $$ Now notice that $2^{-n}\lfloor t2^n \rfloor \leq t$ and $2^{-n}\lfloor t2^n \rfloor \to t$ ($n \to \infty$) and use the left continuity of $X$.