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Please use following results in Billingsley's book: Convergence of Probability Measures, 2ed. p.124 Th3.1 p.27.
Theorem Suppose that $(X_n,Y_n)$ are random elements of $S\times S$. If $X_n\Rightarrow X$ and $\rho(X_n,Y_n)\Rightarrow 0$, then $Y_n\Rightarrow Y$.
Write $$ W_n(t)=M_n(t)+[W_n(t)-M_n(t)], $$ Then i) $M_n\stackrel{w}{\to}W$ in Skorokhod topology;

ii) $\mathsf{P}(\rho(W_n,M_n)>\varepsilon)\to0$, $\forall \varepsilon>0$, since $\mathsf{E}[\sup_t|W_n(t)-M_n(t)|^2]\to 0$ and $\mathsf{P}(\sup_t|W_n(t)-M_n(t)|>\varepsilon)\to0$, $\forall \varepsilon>0$.
Now by using Th.3.1 and from this two facts it is ready to get $W_n\stackrel{w}{\to}W$ in Skorokhod topology.