Let $X$ be separable metric space and $\mathscr{M}\left(X\right)$ be the space of all probability measures on $X$ and $d_{P}$ are Prohorov metric on $\mathscr{M}\left(X\right)$. We denote $\mu_{n}\Rightarrow\mu$ if $\mu_{n}$ converges to $\mu$ weakly. It is well known that
$\mu_{n}\Rightarrow\mu\Longleftrightarrow d_{P}\left(\mu_{n},\mu\right)\rightarrow 0$
However, in many books (like Billingsley (1999).Convergence of Probability Measures), they just conclude that the weak topology on $\mathscr{M}\left(X\right)$ can be induced by $d_{P}$. If the weak topology on $\mathscr{M}\left(X\right)$ is metrizable, we can make sure that this statement is true since topology is determined by convergence in metric space.
But we do not have the result that the weak topology on $\mathscr{M}\left(X\right)$ is metrizable. (I know this statement is true, but I don't know how to prove it.)
So my question is how to prove that the weak topology on $\mathscr{M}\left(X\right)$ can be induced by $d_{P}$? Could anyone help me out? Thanks in advance.