Extension of Slutsky's Theorem

Question

I regard random variables $X_n$ and $Y_n$ with $(X_n+Y_n) \rightarrow (X+Y)$ (in distribution for $n \to \infty$). Furthermore there exist random variables $(a_n) \rightarrow 1$ and $(b_n) \rightarrow 1$ (both in distribution for $n \to \infty$). Does also $(a_nX_n+b_nY_n) \rightarrow (X+Y)$ hold in distribution?

Many thanks in advance.