How exactly is Skorohod's Representation Theorem applied here?

How exactly is Skorohod's Representation Theorem applied here?

I have a question about applying the Skorohod Representation Theorem. Let
$(\Omega,\mathcal{F},Q)$ be a probability space and $F_n\subset
\mathbb{R}_+$ such that $\lim_n F_n = \mathbb{R}_+$. Assume we have a
continuous real-valued function $g:\mathbb{R}_+\to\mathbb{R}$ with
$|g(x)|\le C(1+x^p)$ for $p>2$ and a constant $C$. Let $X$ be a random
variable and define $f_n:=g|_{F_n}$ the restriction to $F_n$. $f_n$ has
the property that $\sup_{x\in F_n}\frac{|f_n(x)|}{(1+x^p)}\le n$ (maybe
this is not needed). Since $g$ is continuous we have that for $x_n\ge 0$,
$x\ge 0$ with $x_n\to x$ then $g(x)=\lim_nf_n(x_n)$.
After all we have a tight sequence $(P_n)$ of probability measures given,
hence there is a subsequence again denoted by $(P_n)$ which converges
weakly to a measure $P$. Now using the Skorohod Representation Theorem we
should establish the following equality:
$$E_P[g(X)]=\lim_nE_{P_n}[f_n(X)]$$
where $E_p[\cdot]$ denotes the expectation w.r.t the measure $P$, and
similaryly for $E_{P_n}[\cdot]$ w.r.t the measure $P_n$. How exactly is in
this setting the Skorohod Repesentation Theorem applied?