Consider a right-continuous super-martingale $(X_u,\mathcal{F}_u)_{u \in \mathbb{R}_+}.$ Let $\theta_1$ and $\theta_2$ be two stopping times such that $\theta_1 \leq \theta_2.$
Prove that $(X_{u\wedge\theta_2}-X_{u\wedge\theta_1},\mathcal{F}_{u})_{u \in \mathbb{R}_+}$ is a super-martingale.
Clearly $X_{u\wedge\theta_2}-X_{u\wedge\theta_1}$ is $\mathcal{F}_u$-measurable and integrable since $E[|X_{u\wedge\theta_1}|] \leq \liminf_{n \to +\infty}E[|X_{\frac{1}{n}\left \lceil{n\theta_1\wedge u}\right \rceil }|] \leq E[X_0]+2\liminf_{n \to +\infty}E[X^-_{\frac{1}{n}\left \lceil{n\theta_1\wedge u}\right \rceil }] \leq E[X_0]+2E[X^-_{\left \lceil{u}\right \rceil +1}]<+\infty$ and $(X^-_{k/n},\mathcal{F}_{k/n})_{k \in \mathbb{N}}$ is a sub-martingale.
We need to prove that for $r \leq u, E[X_{u\wedge\theta_2}-X_{u\wedge\theta_1}|\mathcal{F}_r] \leq X_{r\wedge\theta_2}-X_{r\wedge\theta_1}$ a.s.
Let $q \in \mathbb{N}$ and $n \in \mathbb{N}^*.$ Since $(X_{k/n},\mathcal{F}_{k/n})_{k \in \mathbb{N}}$ is a super-martingale, it follows from the discrete case that $(q \wedge (X_{k \wedge \left \lceil{n\theta_2}\right \rceil/n}-X_{k \wedge \left \lceil{n\theta_1}\right \rceil/n}),\mathcal{F}_{k/n})_{k \in \mathbb{N}}$ is a super-martingale.
Therefore $$E[q \wedge (X_{\left \lceil{n\theta_2}\right \rceil \wedge \left \lceil{nu}\right \rceil/n}-X_{\left \lceil{n\theta_1}\right \rceil \wedge \left \lceil{nu}\right \rceil/n})|\mathcal{F}_{\left \lceil{nr}\right \rceil/n}] \leq q \wedge (X_{\left \lceil{nr}\right \rceil \wedge \left \lceil{n\theta_2}\right \rceil/n}-X_{\left \lceil{nr}\right \rceil\wedge \left \lceil{n\theta_1}\right \rceil/n})$$
We note that $q \wedge (X_{\left \lceil{n\theta_2}\right \rceil \wedge \left \lceil{nu}\right \rceil/n}-X_{\left \lceil{n\theta_1}\right \rceil \wedge \left \lceil{nu}\right \rceil/n}) \leq q.$
How can we provide an lower-bound for $q \wedge (X_{\left \lceil{n\theta_2}\right \rceil \wedge \left \lceil{nu}\right \rceil/n}-X_{\left \lceil{n\theta_1}\right \rceil \wedge \left \lceil{nu}\right \rceil/n})$ of the form $E[Y|\mathcal{F}_{x_n}],$ where $Y$ and $x_n$ to be found so that $q \wedge (X_{\left \lceil{n\theta_2}\right \rceil \wedge \left \lceil{nu}\right \rceil/n}-X_{\left \lceil{n\theta_1}\right \rceil \wedge \left \lceil{nu}\right \rceil/n})$ is uniformly integrable, and later we can take the limit in probability $n \to +\infty$ and to conclude by taking $q \to +\infty.$
Once you know it for discrete supermartingales (which is straightforward), it holds for supermartingales where the time parameter (and the stopping times $\theta_i$) take values in the integer multiples of $2^{-k}$. The general case follows by approximating the stopping times $\theta_i$ from above by $\theta_i(k):=\lceil 2^k \theta_i \rceil2^{-k}$ and passing to the limit as $k \to \infty$ via right continuity.