For $k, n \in \mathbb{N}$ let $\chi_{k}^{n}, \chi_{k}$ be random variables taking values in a metric space $(S,d)$. Suppose the following holds:
For any $k \in \mathbb{N}$ it is the case that:
$(\chi_{1}^{n}, \chi_{2}^{n}, ..., \chi_{k}^{n}) \overset{d}{\to} (\chi_{1}, \chi_{2}, ..., \chi_{k})$ as $n \to \infty$ in the space $S^{k}$ with the standard product 1 norm $d^{k}$
Are there any simple conditions on either $S$ or the random variables under which I can then conclude that:
$(\chi_{1}^{n}, ..., ) \overset{d}{\to} (\chi_{1}, ...)$ in the space of $S$ valued sequence $S^{\infty}$ with the $\sup$ norm.
I don't really have a good idea how to figure this out, I have been trying to think about it using the characterization of distributional convergence in terms of bounded linear functionals, but am having difficulty thinking about the space $C_{B}(S^{\infty}, \mathbb{R})$.
