Given a sequence of stochastic process $\xi_n(t)$ defined for $t \in [t_0,T]$ on ($\Omega,\mathcal{F},\mathbb{P}$) such that its stochastically continuous. Now consider a dense set $t_1,t_2, \cdots,$ in $[t_0,T]$ and assume the existence of a sequence of random variables(they exist due to a deep result due to Skorokhod which I skip for reasons of brevity)on a new probability space ($\Omega',\mathcal{F}',\mathbb{P}'$)
$x_n=\{x_1^{(n)},x_2^{(n)},\dots,\} n=0,1,\dots$ such that $\left(x_1^{(n)},x_2^{(n)},\dots,\right)$ has the same joint distribution as $\left(\xi_n(t_1),\xi_n(t_2), \dots,\right)$ for all finite combinations of $t_i$ in the dense set $t_1,t_2, \cdots,$ of $[t_0,T]$
Clearly the stochastic continuity of $\xi_n(t)$ implies that as $t_{k_r} \to t$ we have that $\xi_n(t_{k_r}) \to \xi_n(t)$ in probability
$\textbf{"Note that in the following $x^{(n)}_I(\omega')$ is a random variable on the space $(\Omega',\mathcal{F}',\mathbb{P}')$ AND NOT THE VALUE taken by the random variable at $\omega'$"}$
Now This is equivalent to $$\lim_{r \to \infty} E\left[\ \frac{\vert\xi_n(t_{k_r})-\xi_n(t)\vert}{1+\vert\xi_n(t_{k_r})-\xi_n(t)\vert}\right]=0$$. If $t=t_k$ for some $k$ in the sequence $t_1,t_2,\cdots,$ then since $\left(\xi_n(t_{k_r}),\xi_n(t)\right)$ has the same joint distribution as $\left(x_{k_r}^n(\omega'),x_k^n(\omega')\right)$, we have that $$\lim_{r \to \infty} E\left[\ \frac{\vert x_{k_r}^n(\omega')-x_k^n(\omega')\vert}{1+\vert x_{k_r}^n(\omega')-x_k^n(\omega')\vert}\right]=0$$ which is equivalent to saying that $x_{k_r}^n \to x_k^n(\omega')$ in probability.
What I CANT SHOW is that of for $t \neq t_k$ why does there exists some random variable say $x^n(t,\omega')$ such that $x_{k_r}^n$ converges to it whenever $t_{k_r} \to t$?
Any help would be highly appreciated. Thanks