}\right\vert \right) \right] \vee
\mathbb{E}\left[ G\left( \left\vert M_{\infty }\right\vert \right) \right]
<\infty .$$Now, define the stopping times $$\tau _{k}^{n}:=\inf \left\{ u>0:\sup_{t\leq u}\left\vert M_{t}^{\left(
n\right) }-M_{t}\right\vert \geq k\right\} .$$Observe that the estimation $\sup_{n\in \mathbb{N}}\mathbb{E}\left[ G\left(
\left\vert M_{\tau _{k}^{n}}^{\left( n\right) }\right\vert \right) \right]
\leq \sup_{n\in \mathbb{N}}\mathbb{E}\left[ G\left( \left\vert M_{\infty
}^{\left( n\right) }\right\vert \right) \right] $ implies the uniformly integrability of $\left\{ M_{\tau _{k}^{n}}^{\left( n\right) }\right\}
_{n\in \mathbb{N}}$ for each $k$ fixed. Since any uniformly integrable càdlàg martingale is of class $\mathcal{D}$, follows the uniform integrability of $\left\{ M_{\tau _{k}^{n}}\right\} _{n\in \mathbb{N}}$ for all $k\in \mathbb{N}$, and hence $\left\{ \sup\nolimits_{t\leq \tau
_{k}^{n}}\left\vert M_{t}^{\left( n\right) }-M_{t}\right\vert \right\}
_{n\in \mathbb{N}}$ is uniformly integrable. This and the maximal inequality for supermartingales $$\begin{aligned}
\mathbb{P}\left[ \sup_{t\in \mathbb{R}_{+}}\left\vert M_{t}^{\left( n\right)
}-M_{t}\right\vert \geq \varepsilon \right] &\leq &\frac{1}{\varepsilon }\left\{ \sup_{t\in \mathbb{R}_{+}}\mathbb{E}\left[ \left\vert M_{t}^{\left(
n\right) }-M_{t}\right\vert \right] \right\} \\
&\leq &\frac{1}{\varepsilon }\mathbb{E}\left[ \left\vert M_{\infty }^{\left(
n\right) }-M_{\infty }\right\vert \right] \longrightarrow 0,\end{aligned}$$yields the convergence of $\left\{ \sup\nolimits_{t\leq \tau
_{k}^{n}}\left\vert M_{t}^{\left( n\right) }-M_{t}\right\vert \right\}
_{n\in \mathbb{N}}$ in $L^{1}$ to $0$. The second Davis’ inequality [@HeWanYan Thm. 10.28] guarantees that, for some constant $C$, $$\mathbb{E}\left[ \sqrt{\left[ M^{\left( n\right) }-M\right] _{\tau _{k}^{n}}}\right] \leq C\mathbb{E}\left[ \sup\limits_{t\leq \tau _{k}^{n}}\left\vert
M_{t}^{\left( n\right) }-M_{t}\right\vert \right] \underset{n\rightarrow
\infty }{\longrightarrow }0\quad \forall 