I have been reading the Karoui's paper about Reflected Backward Stochastic Diffrential Equation.
Let $b:[0,T]\times\mathbb{R}^d\rightarrow\mathbb{R}^d$ and $\sigma:[0,T]\times\mathbb{R}^d\rightarrow\mathbb{R}^{d\times d}$ be continuous mappings, which are Lipschitz with respect to their second variable, uniformly with respect to $t\in[0,T]$. For each $(t,x)\in[0,T]\times\mathbb{R}^d$, let $\left\{X_{s}^{t,x}; t\leq s\leq T\right\}$ be the unique $\mathbb{R}^d$-valued solution of the SDE: $$X_{s}^{t,x}=x+\int_{t}^{s}b(r,X_{r}^{t,x})dr+\int_{t}^{s}\sigma(r,X_{r}^{t,x})dBr.$$ On page number 729, they say that $$\begin{align*} \mathbb{E}(\left|g(X_T^{t,x} )-g(X_T^{t_n,x_n})\right|^{2})&\rightarrow 0 \\ \mathbb{E}(\displaystyle\sup_{0\leq s\leq T}\left|h(s,X_s^{t,x} )-h(s,X_s^{t_n,x_n})\right|^{2})&\rightarrow 0, \end{align*}$$
where $g\in C(\mathbb{R}^d)$ and has most polynomial growth at infinity and $h:[0,T]\times\mathbb{R}^d\rightarrow\mathbb{R}$ is jointly continuous in $t$ and $x$ and satisfies
$$h(t,x)\leq K(1+\left|x\right|^p), \hspace{1cm} t\in[0,T], x\in\mathbb{R}^d$$
and
$$h(T,x)\leq g(x), \hspace{0.5cm} x\in\mathbb{R}.$$
So, I think that the first convergence is thanks to dominated convergence theorem (DCT), since for $\omega\in \Omega$ we have $g(X_T^{t_n,x_n}(\omega))\rightarrow g(X_{T}^{t,x}(\omega))$ by the continuity of $g$, $X_T^{t,x}$ (respect to the initial conditions $t$ and $x$) and
$\mathbb{E}(|g(X_T^{t,x})|^2)\leq\mathbb{E}(C(1+|X_T^{t,x}|^2))\leq C(1+|x|^2)<\infty$.
Am I right or wrong? if I am right, can I use the same idea for the second convergence? or, is the supremum a problem in order to do it ?
Thank you in advance!